ji_n_.  _n__n_ ry 


REESE  LIBRARY 

OF  THK 


UNIVERSITY  OF  CALIFORNIA. 


,190 
.   Class  No. 


Accession  No. 

St—u—u—v—u-'U—u—u—u-v—u    u    u     n'"ir    u—  u-u—  u 


|Q 

<^ 


Theoretical    Elements 


of 


Electrical  Engineering 


By 

Charles    Proteus    Steinmetz 


.,      .  ;!TY 


OF 


New  York : 
Electrical  World    and    Engineer 

(Incorporated) 


fK 


COFYRIGHTEDj  IQOI,  BY 

ELECTRICAL  WORLD   AND   ENGINEER 
NEW  YORK 


TYPOGRAPHY  BY  C.  J.  PETERS  A  SON 
BOSTON,  U.S.A. 


J 


PREFACE. 


THE  first  part  of  the  following  volume  originated  from  a 
series  of  University  lectures  which  I  once  promised  to  deliver. 
This  part  can,  to  a  certain  extent,  be  considered  as  an  in- 
troduction to  my  work  on  "  Theory  and  Calculation  of 
Alternating  Current  Phenomena,"  leading  up  very  gradually 
from  the  ordinary  sine  wave  representation  of  the  alternating 
current  to  the  graphical  representation  by  polar  coordinates, 
from  there  to  rectangular  components  of  polar  vectors,  and 
ultimately  to  the  symbolic  representation  by  the  complex 
quantity.  The  present  work  is,  however,  broader  in  its 
scope,  in  so  far  as  it  comprises  the  fundamental  principles 
not  only  of  alternating,  but  also  of  direct  currents. 

The  second  part  is  a  series  of  monographs  of  the  more 
important  electrical  apparatus,  alternating  as  well  as  direct 
current.  It  is,  in  a  certain  respect,  supplementary  to  "Al- 
ternating Current  Phenomena."  While  in  the  latter  work  I 
have  presented  the  general  principles  of  alternating  current 
phenomena,  in  the  present  volume  I  intended  to  give  a 
specific  discussion  of  the  particular  features  of  individual 
apparatus.  In  consequence  thereof,  this  part  of  the  book 
is  somewhat  less  theoretical,  and  more  descriptive,  my  inten- 
tion being  to  present  the  most  important  electrical  apparatus 
in  all  their  characteristic  features  as  regard  to  external  and 
internal  structure,  action  under  normal  and  abnormal  condi- 
tions, individually  and  in  connection  with  other  apparatus,  etc. 

I  have  restricted  the  work  to  those  apparatus  which  ex- 
perience has  shown  as  of  practical  importance,  and  give  only 


9101 


iv  PREFACE. 

those  theories  and  methods  which  an  extended  experience 
in  the  design  and  operation  has  shown  as  of  practical  utility. 
I  consider  this  the  more  desirable  as,  especially  of  late  years, 
electrical  literature  has  been  haunted  by  so  many  theories 
(for  instance  of  the  induction  machine)  which  are  incorrect, 
or  too  complicated  for  use,  or  valueless  in  practical  applica- 
tion. In  the  class  last  mentioned  are  most  of  the  graphical 
methods,  which,  while  they  may  give  an  approximate  insight 
in  the  inter-relation  of  phenomena,  fail  entirely  in  engineering 
practice  owing  to  the  great  difference  in  the  magnitudes  of 
the  vectors  in  the  same  diagram,  and  to  the  synthetic  method 
of  graphical  representation,  which  generally  requires  one 
to  start  with  the  quantity  which  the  diagram  is  intended  to 
determine. 

I  originally  intended  to  add  a  chapter  on  Rectifying  Ap- 
paratus, as  arc  light  machines  and  alternating  current  recti- 
fiers, but  had  to  postpone  this,  due  to  the  incomplete  state 
of  the  theory  of  these  apparatus. 

The  same  notation  has  been  used  as  in  the  Third  Edition 
of  "Alternating  Current  Phenomena,"  that  is,  vector  quanti- 
ties denoted  by  dotted  capitals.  The  same  classification 
and  nomenclature  have  been  used  as  given  by  the  report  of 
the  Standardizing  Committee  of  the  American  Institute  of 
Electrical  Engineers. 


CONTENTS. 


PART   I. 
GENERAL  THEORY. 

PAGE 

1.  Magnetism  and  Electric  Current 1 

4  examples 5 

2.  Magnetism  and  E.M.F. 9 

2  examples 10 

3.  Induction  of  E.M.F's 11 

3  examples 14 

4.  Effect  and  Effective  Values 15 

3  examples 16 

5.  Self-Induction  and  Mutual  Induction     .     .     .     . 18 

2  examples .21 

6.  Self-induction  of  Continuous  Current  Circuits 22 

5  examples 26 

7.  Self-induction  in  Alternating  Current  Circuits 29 

2  examples 34 

8.  Effect  of  Alternating  Currents .  38 

1  example 41 

9.  Polar  Coordinates 41- 

1  example 45 

10.  Hysteresis  and  Effective  Resistance 48 

1  example 53 

11.  Capacity  and  Condensers 55 

1  example "8 

12.  Impedance  of  Transmission  Lines 58 

2  examples 05 

13.  Alternating  Current  Transformer  .     .  _ 09 

1  example 77 

14.  Rectangular  Coordinates 79 

3  examples 85 

15.  Load  Characteristic  of  Transmission  Line 89 

1  example 93 

16.  Phase  Control  of  Transmission  Lines 94 

2  examples 99 

v 


VI  CONTENTS. 

PAGE 

17.  Impedance  and  Admittance 103 

2  examples 108 

18.  Equivalent  Sine  Waves 112 

1  example 114 


PART   II. 
SPECIAL  APPARATUS. 

INTRODUCTION.    Classification  and  Nomenclature  of  Apparatus  121 

A.  Synchronous  Machines 126-101 

I.   General 126 

II.   E.M.F.'s 128 

III.   Armature  Reaction 129 

IV.   Self-Induction 133 

V.   Synchronous  Reactance 136 

VI.    Characteristic  Curves  of  Alternating  Current  Generator    .  139 

VII.    Synchronous  Motor 141 

VIII.   Characteristic  Curves  of  Synchronous  Motor 143 

IX.    Magnetic  Characteristic  or  Saturation  Curve 147 

X.   Efficiency  and  Losses lt"0 

XI.   Unbalancing  of  Polyphase  Synchronous  Machines    .     .    .  151 

XII.   Starting  of  Synchronous  Motors    ..." 153 

XIII.  Parallel  Operation 154 

XIV.  Division  of  Load  in  Parallel  Operation 156 

XV.    Fluctuating  Cross-Currents  in  Parallel  Operation  ....  157 

B.  Commutating  Machines   ...  * 162-216 

I.  General \    .     .  162 

II.  Armature  Winding 165 

III.  Induced  E.M.F.'s 175 

IV.  Distribution  of  Magnetic  Flux 177 

V.   Effect  of  Saturation  on  Magnetic  Distribution 181 

VI.   Effect  of  Slots  on  Magnetic  Flux 184 

VII.   Armature  Reaction 187 

VIII.    Saturation  Curves 188 

IX.    Compounding 190 

X.   Characteristic  Curves 192 

XI.   Efficiency  and  Losses 193 

XII.   Commutation *193 

XIII.   Types  of  Commutating  Machines 202 

A.  Generators.      Separately  excited    and   Magneto, 
Shunt,  Series,  Compound 205 

B.  Motors.     Shunt,  Series,  Compound 212 


CONTENTS.  vii 

PAGB 

C.  Synchronous  Converters 217-260 

I.    General 217 

II.    Ratio  of  E.M.F.'s  and  of  Currents 218 

III.  Variation  of  the  Ratio  of  E.M.F.'s 225 

IV.  Armature  Current  and  Heating 228 

V.   Armature  Reaction 286 

VI.    Wattless  Currents  and  Compounding 242 

VII.    Starting 244 

VIII.   Inverted  Converters 240 

IX.   Double  Current  Generators 248 

X.    Conclusion    .     .• , 250 

APPENDIX.     Direct  Current  Converter 251 

D.  Induction  Machines 261-320 

I.    General ,     .     .     .     .  261 

II.    Polyphase  Induction  Motor. 

1.  Introduction 2(>5 

2.  Calculation 26(5 

3.  Load  and  Speed  Curves 272 

4.  Effect  of  Armature  Reaction  and  Starting.     .     .     .  277 

III.  Single-phase  Induction  Motor. 

1.  Introduction 281 

2.  Load  and  Speed  Curves 286 

3.  Starting  Devices  of  Single-phase  Motors    ....  21)0 

4.  Acceleration  with  Starting  Device 295 

IV.  Induction  Generator. 

1.  Introduction 297 

2.  Constant  Speed   Induction  or  Asynchronous  Gene- 

rator         299 

3.  Power  Factor  of  Induction  Generator 301 

V.   Induction  Booster 307 

VI.    Phase  Converter 309 

VII.   Frequency   Converter    or    General    Alternating    Current 

Transformer 312 

VIII.   Concatenation  of  Induction  Motors  .                                   ,  315 


PART  L 

GENERAL    THEORY, 


1.    MAGNETISM    AND    ELECTRIC    CURRENT. 

A  magnet  pole  attracting  (or  repelling)  another  magnet 
pole  of  equal  strength  at  unit  distance  with  unit  force  *  is 
called  a  unit  magnet  pole. 

The  space  surrounding  a  magnet  pole  is  called  a  mag- 
netic field  of  force,  or  magnetic  field. 

The  magnetic  field  at  unit  distance  from  a  unit  magnet 
pole  is  called  a  unit  magnetic  field,  and  is  represented  by 
one  line  of  magnetic  force  (or  shortly  "one  line")  per  cm2, 
and  from  a  unit  magnet  pole  thus  issue  a  total  of  4  TT  lines 
of  magnetic  force. 

The  total  number  of  lines  of  force  issuing  from  a  magnet 
pole  is  called  its  magnetic  flux. 

The  magnetic  flux  <£  of  a  magnet  pole  of  strength  m  is, 


At  the  distance  R  from  a  magnet  pole  of  strength  ;«, 
and  therefore  of  flux  <|>  =  4ir/«,  the  magnetic  field  has  the 
intensity, 


since  the  <I>  lines  issuing  from  the  pole  distribute  over  the 
area  of  a  sphere  of  radius  R,  that  is  the  area  4  -n-R2. 

*  That  is,  at  1  cm  distance  with  such  force  as  to  give  to  the  mass  of  1  gramme  the 
acceleration  of  1  cm  per  second. 


ELECTRICAL   ENGINEERING 


A  magnetic  field  of  intensity  X  exerts  upon  a  magnet 
pole  of  strength  m  the  force, 


Thus  two  magnet  poles  of  strengths  ml  and  -mv  and  dis- 
tance R,  exert  upon  each  other  the  force, 


Electric  currents  produce  magnetic  fields  also.  That  is, 
the  space  surrounding  the  conductor  carrying  an  electric 
current  is  a  magnetic  field,  which  appears  and  disappears 
and  varies  with  the  current  producing  it,  and  is  indeed  an 
essential  part  of  the  phenomenon  called  an  electric  current. 

Thus  an  electric  current  represents  a  magnetomotive 
force  (M.M.F.). 

The  magnetic  field  of  a  straight  conductor  consists  of 
lines  of  magnetic  force  surrounding  the  conductor  in  con- 
centric circles.  The  intensity  of  this  magnetic  field  is  pro- 
portional to  the  current  strength  and  inversely  proportional 
to  the  distance  from  the  conductor. 

Unit  current  is  the  current  which  in  a  straight  conductor 
produces  unit  field  intensity  at  unit  distance  from  the  con- 
ductor, that  is,  one  .line  per  cm2  in  the  magnetic  circuit  of 
2  TT  cm.  length  surrounding  the  conductor  as  concentric 
circle,  or  twice  this  field  intensity  at  unit  distance  from  a 
closed  current  loop,  that  is  a  turn,  consisting  of  conductor 
and  return  conductor. 

One-tenth  of  unit  current  is  the  practical  unit,  called  one 
ampere. 

One  ampere-turn  thus  produces  at  unit  distance  from  the 
conductor  the  field  intensity  .2,  and  at  distance  R  the  field 

.2 
intensity  -=  ,  and  SF  ampere-turns  the  field  intensities  3C  =  .  2  <$ 

.2& 

and  X  =  —  —  ,  respectively. 

SF,  that  is,  the  product  of  amperes  and  turns,  is  called 
magnetomotive  force  (M.M.F.). 


MAGNETISM  AND   ELECTRIC   CURRENT.  3 

The  M.M.F.  per  unit  length  of  magnetic  circuit,  or  ratio, 

M.M.F. 

*       length  of  magnetic  circuit 

is  called  the  magnetizing  force. 

Thus,  at  unit  distance  from  the  conductor  of  a  loop  of 
&  ampere-turns  M.M.F.,  the  magnetizing  force  is 


or  at  distance  R 

/= 


and  since  the  field  intensity  at  distance  R  is 
TC     '2(F 

'-R 

we  have, 


If  a  conductor  is  coiled  in  a  spiral  of  /  cm  length  and  N 

N 
turns,   thus  «  =  -7  turns   per  cm   length  of    spiral,  and  / 

=  current  in  amperes  passing  through  the  conductor,  the 
M.M.F.  of  the  spiral  is, 

$  =  NI 

and  the  magnetizing  force  in  the  middle  of  the  spiral  'that 
is,  neglecting  the  reaction  of  the  ends), 


Thus,  the  field  intensity  in  the  middle  of  the  spiral  or 

solenoid, 

OC  =  .4  TT 


That  is,  1  ampere-turn  per  cm  length  of  magnetic  circuit 
produces  .4  -n-  =  1.257  lines  of  magnetic  force  per  cm2. 

10  amperes,  or  unit  current,  per  cm  length  of  magnetic 
circuit,  produces  4  TT  lines  of  magnetic  force  per  cm2.  That 


4  ELECTRICAL   ENGINEERING. 

is,  unit  current  is  the  current  which,  when  acting  upon  a 
magnetic  circuit  (in  air)  of  unit  length  (or  1  cm),  produces 
the  same  number  of  lines  of  magnetic  force  per  cm2,  (4  TT), 
as  issue  altogether  from  a  unit  magnet  pole. 

M.M.F.  $  applies  to  the  total  magnetic  circuit,  or  part  of 
the  magnetic  circuit.     It  is  measured  in  ampere-turns. 

Magnetizing  force  f  is  the  M.M.F.  per  unit  length  of 
magnetic  circuit.     It  is  measured  in  ampere-turns  per  cm. 

Field  intensity  JC  is  the  number  of  lines  of  force  per  cm2. 
If  /  =  length  of   magnetic  circuit  or  part   of  magnetic 
circuit, 

fF  =  //  /=* 


=  1.257/  /=.796JC. 

The  preceding  applies  only  to  magnetic  fields  in  air  or 
other  unmagnetic  materials. 

If  the  medium  in  which  the  magnetic  field  is  established 
is  a  "  magnetic  material,"  the  number  of  lines  of  force  per 
cm2  is  different  and  usually  many  times  greater.  (Slightly 
less  in  diamagnetic.  materials.) 

The  ratio  of  the  number  of  lines  of  force  in  a  medium, 
to  the  number  of  lines  of  force  which  the  same  magnetizing 
force  would  produce  in  air  (or  rather  in  a  vacuum),  is  called 
the  permeability  or  magnetic  conductivity  /u,  of  the  medium. 

The  number  of  lines  of  force  per  cm2  in  a  magnetic 
medium  is  called  the  magnetic  induction  <B.  The  number 
of  lines  of  force  produced  by  the  same  magnetizing  force  in 
air  is  called  the.  field  intensity  3C. 

In  air,  magnetic  induction  (B  and  field  intensity  3C  are 
equal. 

As  a  rule,  the  magnetizing  force  in  a  magnetic  circuit  is 
changed  by  the  introduction  of  a  magnetic  material,  due  to 
the  change  of  distribution  of  the  magnetic  flux. 

The  permeability  of  air  =  1  and  is  constant 


MAGNETISM  AND  ELECTRIC   CURRENT.  5 

The  permeability  of  iron  and  other  magnetic  materials 
varies  with  the  magnetizing  force  between  a  little  above  1 
and  about  4000  in  soft  iron. 

The  magnetizing  force/  in  a  medium  of  permeability  /x 
produces  the  field  intensity  3C  =  .4  irf  and  the  magnetic 
induction  &  =  .4  TT/*/". 

EXAMPLES. 

(1.)  To  hold  a  horizontal  bar  magnet  of  12  cm  length, 
pivoted  in  its  center,  in  a  position  at  right  angles  to  the 
magnetic  meridian,  a  pull  is  required  of  2  grams  at  4  cm 
radius.  What  is  the  intensity  of  the  poles  of  the  magnet, 
and  the  number  of  lines  of  magnetic  force  issuing  from  each 
pole,  if  the  horizontal  intensity  of  the  terrestrial  magnetic 
field  OC  =  .2,  and  the  acceleration  of  gravity  =  980  ? 

As  distance  of  the  poles  of  the  bar  magnet  may  be 
assumed  £  of  its  length. 

Let  m  =  intensity  of  magnet  pole,  r  =  5  is  the  radius 
on  which  the  terrestrial  magnetism  acts. 

Thus  2m3C,r=  2m  =  torque  exerted  by  the  terrestrial 
magnetism. 

2  grams  weight  =  2  x  980  =  1960  units  of  force.  These 
at  4  cm  radius  give  the  torque  4  X  1960  =  7840. 

Hence  2  m  =  7840. 

m  =  3920  is  the  strength  of  each  magnet  pole  and 

<£  =  4  tnn  =  49000,  the  number  of  lines  of  force  issuing 
from  each  pole. 

(2.)  A  conductor  carrying  100  amperes  runs  in  the  direc- 
tion of  the  magnetic  meridian.  What  position  will  a  com- 
pass needle  assume,  when  held  vertically  below  the  conductor 
at  a  distance  of  50  cm,  if  the  intensity  of  the  terrestrial 
magnetic  field  is  .2  ? 

The  intensity  of  the  magnetic  field  of  100  amperes  50  cm 

.27  100 

from  the  conductor  is  3C  =  —^-  =  .2  X  -=^-  =  .4,  the  direc- 

A  OU 

tion  is  at  right   angles  to  the  conductor,  that  is  at   right 
angles  to  the  terrestrial  magnetic  field. 


6  ELECTRICAL   ENGINEERING. 

If  <£  =  angle  between  compass  needle  and  the  north  pole 
of  magnetic  meridian,  /  =  length  of  needle,  m  =  intensity 
of  its  magnet  pole,  the  torque  of  the  terrestrial  magnetism 
is  3£ml  sin  <£  =  .2  ml  sin  <£,  the  torque  of  the  current  is 

.  2  Im  I  cos  <£ 

„        -  =  .4  ;/z/ cos  <£. 
K 

In  equilibrium,  .2  ml  sin  <£  =  .4  ml  cos  <£,  or  tan  <£  =  2, 
<£  =  63.4°. 

(3.)  What  is  the  total  magnetic  flux  per  /=1000  m 
length,  passing  between  the  conductors  of  a  long  distance 
transmission  line  carrying  /  amperes  of  current,  if  d=  .82 
cm  is  the  diameter  of  the  conductors  (No.  0  B.  &  S.  G.), 
D  =  45  cm  their  distance  from  each  other  ? 


Fig.  1. 

At  distance  x  from  the  center  of  one  of  the  conductors 
(Fig.  1),  the  length  of  the  magnetic  circuit'  surrounding  this 
conductor  is  2  irx,  the  M.M.F.  7 ampere  turns  ;  thus  the  mag- 
netizing force  f  —  T) —  >  and  the  field  intensity  3C  =  .4  vf 

.27  .Zlldx 

=  — ,  and  the  flux  in  the  zone  dx  is  d$  = ,  and 

x  x 

the  total  flux  from  the  surface  of  the  conductor  to  the  next 
conductor  is,  D 

r  .siid* 

<£  =   I       = 

,  J   d  X 

2Z> 


MAGNETISM  AND   ELECTRIC   CURRENT.  7 

The  same  flux  is  produced  by  the  return  conductor  in 
the  same  direction,  thus  the  total  flux  passing  between  the 
transmission  wires  is, 

2  $  =  .4  //  loge  ?jr 

or  per  1000  m  =  105  cm  length, 

QO 

2  *  =  .4  x  105/loge  -^  -  .4  x  105  x  4.70  =  .188  x  106/, 
.o/ 

or  .188  /  megalines  or  millions  of  lines  per  line  of  1000  m 
of  which  .094  /  megalines  surround  each  of  the  two  con- 
ductors. 

J2o  A>*&- 
(4.)    In  an  alternator  each  pole  has  to  carry  6.4  millions 

of  lines,  or  6.4  ml  (megalines)  magnetic  flux.  How  many 
ampere  turns  per  pole  are  required  to  produce  this  flux,  if 
the  magnetic  circuit  in  the  armature  of  laminated  iron  has 
the  cross  section  of  930  cm2  and  the  length  of  15  cm,  the 
air-gap  between  stationary  field  and  revolving  armature  has 
»95  cm  length  and  1200  cm2  section,  the  field-pole  has  26.3 
cm  length  and  1075  cm2  section,  of  laminated  iron,  and  the 
outside  return  circuit  or  yoke  has  a  length  per  pole  of  20 
cm  and  2250  cm2  section,  of  cast  iron? 
The  magnetic  densities  are, 

In  armature  (Ex  =  6880 
In  air-gap  <&2  =  5340 
In  field-pole  (E3  =  5950 
In  yoke  (B4  =  2850 

The  permeability  of  sheet  iron  is  ^  =  2550  at  (^  =  6880, 
^  =  2300  at  (&3  =  5950.  The  permeability  of  cast  iron  is 

^  =  280  at  (B4  =  2850. 

/         fR\ 
Thus  the  field  intensity  IK  =-^lis,  3^  =  2.7,  OC2  =  5340, 

OC3  =  2.5,  3C4  =  10.2. 

The  magnetizing  force  (f=  ~\  is,  /x  =  2.15,  /2  =  4250, 

/3  =  1.99,/4  =  8.13  ampere-turns  per  cm. 

Thus  the  M.M.F.  ($  =//)  is,  ^  =  32,  JF2  =  4030,  <F3 
=  52,  ?F4  =  163, 


8 


ELECTRICAL   ENGINEERING. 


or  the  total  M.M.F.  per  pole, 

^  =  &!  +  $2  H-  $3  +  $4  =  4280  ampere-turns. 

The  permeability  /u,  of  magnetic  materials  varies  with  the 
density  (B,  thus  tables  have  to  be  used  for  these  quantities. 
Such  tables  are  usually  made  out  for  density  (B  and  mag- 
netizing force  /,  so  that  the  magnetizing  force  f  correspond^ 
ing  to  the  density  (B  can  directly  be  derived  from  the  table. 
Such  a  table  is  given  in  Fig.  2. 


50    100     150    200     250    300    350      400    450    500    550     600     650    700    750   800 


lagmrtizing  Forct  f  Anipmturns  jer  ct  i.  Lei  gth  or  Magn«t5c  Cireu 


25     -30      35       40     45       50      55      60      60       JO       75     80 


MAGNETISM  AND  E.M.F. 


2.    MAGNETISM    AND    E.M.F. 

In  an  electric  conductor  moving  relatively  to  a  magnetic 
field,  an  E.M.F.  is  induced  proportional  to  the  rate  of  cutting 
of  the  lines  of  magnetic  force  by  the  conductor.  . 

Unit  E.M.F.  is  the  E.M.F.  induced  in  a  conductor  cutting 
one  line  of  magnetic  force  per  second. 

108  times  unit  E.M.F.  is  the  practical  unit,  called   the 
wit. 

Coiling  the  conductor  n  fold  increases  the  E.M.F.  n  fold, 
by  cutting  each  line  of  magnetic  force  ;/  times. 

In  a  closed  electric  circuit  the  E.M.F.  produces  an 
electric  current. 

The  ratio  of  E.M.F.  to  electric  current  produced  thereby 
is  called  the  resistance  of  the  electric  circuit. 

Unit  resistance  is  the  resistance  of   a  circuit  in  which 

% 

unit  E.M.F.  produces  unit  current. 

109  times  unit  resistance  is  the  practical  unit,  called  the 
ohm. 

The  ohm  is  the  resistance  of  a  circuit,  in  which  one  volt 
produces  one  ampere. 

The  resistance  per  unit  length  and  unit  section  of  a  con- 
ductor is  called  its  resistivity,  p. 

The  resistivity  p  is  a  constant  of  the  material,  varying 
with  the  temperature. 

The  resistance  r  of  a  conductor  of  length  /,  section  s,  and 

lp 
resistivity  p  is  r  =  — 

If  the  current  in  the  electric  circuit  changes,  starts,  or 
stops,  the  corresponding  change  of  the  magnetic  field  of  the 
current  induces  an  E.M.F.  in  the  conductor  conveying  the 
current,  which  is  called  the  E.M.F.  of  self -induction. 

If  the  E.M.F.  in  an  electric  circuit  moving  relatively  to 
a  magnetic  field  produces  a  current  in  the  circuit,  the  mag- 
netic field  produced  by  this  current  is  called  its  magnetic 
reaction. 


10 


ELECTRICAL    ENGINEERING. 


The  fundamental  law  of  self-induction  and  magnetic 
reaction  is,  that  these  effects  take  place  in  such  a  direction 
as  to  oppose  their  cause  (Lentz's  Law). 

Thus  the  E.M.F.  of  self-induction  during  a  rise  of  current 
is  in  the  opposite  direction,  during  a  decrease  of  current  in 
the  same  direction  as  the  E.M.F.  producing  the  current. 

The  magnetic  reaction  of  the  current  induced  in  a  circuit 
moving  out  of  a  magnetic  field  is  in  the  same  direction  in 
a  circuit  moving  into  a  magnetic  field  in  opposite  direction 
to  the  magnetic  field. 

Essentially,  this  law  is  nothing  but  a  conclusion  from  the 
law  of  conservation  of  energy. 

EXAMPLES. 

(1.)  An  electro  magnet  is  placed  so  that  one  pole  sur- 
rounds the  other  pole  cylindrically  as  shown  in  Fig.  3,  and 


Fig.  3. 

a  copper  cylinder  revolves  between  these  poles  with  3000 
revolutions  per  minute.  What  is  the  E.M.F.  induced  be- 
tween the  ends  of  this  cylinder,  if  the  magnetic  flux  of  the 
electro  magnet  is  ®  =  25  ml  ? 

During  each  revolution  the  copper  cylinder  cuts  25  ml. 
It  makes  50  revolutions  per  second.  Thus  it  cuts  50  x  25 
X  106  =  12.5  X  108  lines  of  magnetic  force  per  second. 
Hence  the  induced  E.M.F.  is  E  =  12.5  volts. 


INDUCTION  OF  E.M.FIS.  11 

(This  is  called  "unipolar,"  or  more  properly  "nonpolar" 
induction.) 

(2.)  The  field  spools  of  the  20  polar  alternator  in  sec- 
tion 1,  example  4,  are  wound  each  with  616  turns  of  .106 
cm2  section  (No.  7  B.  &  S.  G.)  and  160  cm  mean  length  of 
turn.  The  20  spools  are  connected  in  series.  How  many 
amperes  and  how  many  volts  are  required  for  the  excita- 
tion of  this  alternator  field,  if  the  resistivity  of  copper  is 
1.8  X  10-6? 

Since  616  turns  on  each  field  spool  are  used,  and  4280 

4-^SO 
ampere^turns  required,  the  current  is  -rrrrr  =  6.95  amperes. 

The  resistance  of  20  spools  of  616  turns  of  160  cm  length, 
.106  cm2  section,  and  1.8  x  10~6  resistivity  is, 

20  x  616  x  160  x  1.8  x  10-6 

—  -  =  33.2  ohms, 

.lUo 

and  the  E.M.F.  required  6.95  x  33.2  =  230  volts. 


3.    INDUCTION    OF    E.M.F.'S. 

A  closed  conductor,  convolution  or  turn,  revolving  in  'a 
magnetic  field,  passes  during  each  revolution  through  two 
positions  of  maximum  inclosure  of  lines  of  magnetic  force 
A  in  Fig.  4,  and  two  positions  of  zero  inclosure  of  lines  of 
magnetic  force  B  in  Fig.  4. 


Fig.  4. 


Thus  it  cuts  during  each  revolution  four  times  the  lines 
of  force  inclosed  in  the  position  of  maximum  inclosure. 


12  ELECTRICAL   ENGINEERING. 

If  $  =  maximum  number  of  lines  of  force  inclosed  by 

'the   conductor,    N  —  number    of    complete    revolutions    per 

second  or  cycles,  and  n  =  number  of  convolutions  or  turns 

of  the  conductor,  the  lines  of  force  cut  per  second  by  the 

conductor,  and  thus  the  average  induced  E-.M.F.  is, 

E  =  4  NnQ  absolute  units, 
volts. 


If  N  is  given  in  hundreds  of  cycles,  <£  in  ml, 
E  =  4  JVn3?  volts. 

If  a  coil  revolves  with  uniform  velocity  through  a  uni- 
form magnetic  field,  the  magnetism  inclosed  by  the  coil  is, 

<l>  cos  <f> 

where  <£  =  maximum  magnetism  inclosed   by  the  coil  and 

fj>  =  angle  between  coil  and 
its  position  of  maximum  in- 
closure  of  magnetism.  (Fig. 
5.) 

&cos0  The  E.M.F.  induced  in 
the  coil,  which  varies  with 
the  rate  of  cutting  or  change 


ng.  5.  of  <£  cos  </>,  is  thus, 

e  =  EQ  sin  <£ 

where  JEQ  is  the  maximum  value  of    E.M.F.,  which   takes 
place  for  <j>  =  90°,  or  at  the  position  of   zero  inclosure  of 

magnetic  flux. 

2 
Since  Avg.  (sin  <f>)  =  -,  the  average  induced  E.M.F.  is, 

7T  ' 

2 

E    =    -  EQ.-T* 

7T 

Since,  however,  we  found  above, 

E  =  47V>2<$>  is  the  average  induced  E.M.F., 
it  follows  that 


is  the  maximum,  and 
=  2  irNnQ  sin  <    the  instantaneous  induced  E.M.F. 


INDUCTION  OF  E.M.F: 'S.  13 

With  uniform  rotation  the  angle  <j>  is  proportional  to  the 
time  /,  0  =  2  TT,  giving  7—  1  /  ^  the  time  of  one  complete 
period,  or  revolution  in  a  bipolar  field,  or  1  //  of  one  revo- 
lution in  a  2/  polar  field. 

Thus, '  <£  =  2-jrNt 

e  =  2  TrNnQ?  sin  2  irNt. 

If  the  time  is  not  counted  from  the  moment  of  maximum 
inclosure  of  magnetic  flux,  but  ^  =  the  time  at  this  moment, 
we  have 

e  =  2-7rNn3?  sin  2 
or,  e  =  2  TrNn®  sin  (<j,  -  ^i;. 

Where  ^  =  2  IT A^  is  the  angle  at  which  the  position  of 
maximum  inclosure  of  magnetic  flux  takes  place,  and  is 
called  its  phase. 

These  E.M.F.'s  are  alternating. 

If  at  the  moment  of  reversal  the  connections  between 
the  coil  and  the  external  circuit  are  reversed,  the  E.M.F.  in 
the  external  circuit  is  pulsating  between  zero  and  £0,  but 
has  the  same  average  value  E. 

If  a  number  of  coils  connected  in  series  follow  each  other 
successively  in  their  rotation  through  the  magnetic  field,  as 
the  armature  coils  of  a  direct  current  machine,  and  the  con- 
nections of  each  coil  with  the  external  circuit  are  reversed 
at  the  moment  of  reversal  of  its  E.M.F.,  their  pulsating 
E.M.F.'s  superimposed  in  the  external  circuit  make  a  more 
or  less  steady  or  continuous  external  E.M.F. 

The  average  value  of  this  E.M.F.  is  the  sum  of  the  aver- 
age values  of  the  E.M.F.'s  of  the  individual  coils. 

Thus  in  a  direct  current  machine,  if  $  =  maximum  flux 
inclosed  per  turn,  ;/  =  total  number  of  turns  in  series  from 
commutator  brush  to  brush,  and  N=  frequency  of  rotation 
through  the  magnetic  field. 

E  =  4  Nn <£  =  induced    E.M.F.   (<J>   in    megalines,   N  in 
hundreds  of  cycles  per  second). 

This  is  the  formula  of  direct  current  induction. 


14  ELECTRICAL   ENGINEERING. 


EXAMPLES. 

(1.)  A  circular  wire  coil  of  200  turns  and  40  cm  mean 
diameter  is  revolved  around  a  vertical  axis.  What  is  the 
horizontal  intensity  of  the  magnetic  field  of  the  earth,  if  at 
a  speed  of  900  revolutions  per  minute  the  average  E.M.F. 
induced  in  the  coil  is  .028  volts  ? 

4027T 

The  mean  area  of-  the  coil  is      ,      =  1255  cm2,  thus  the 

terrestrial  flux  inclosed  is  1255  3C,  and  at  900  revolutions 
per  minute  or  15  revolutions  per  second,  this  flux  is  cut 
4  X  15  =  60  times  per  second  by  each  turn,  or  200  X  60 
=  12000  times  by  the  coil.  Thus  the  total  number  of  lines 
of  magnetic  force  cut  by  the  conductor  per  second  is  12000 
;x  1255  OC  =  .151  X  108OC,  and  the  average  induced  E.M.F. 
is  .151  OC  volts.  Since  this  is  =  .028  volts,  X  =  .186. 

(2.)  In  a  550  volt  direct  current  machine  of  8  poles  and 
500  revolutions,  with  drum  armature,  the  average  voltage 
per  commutator  segment  shall  not  exceed  11,  each  arma- 
ture coil  shall  contain  one  turn  only,  and  the  number  of 
commutator  segments  per  pole  shall  be  divisible  by  3,  so  as 
to  use  the  machine  as  three-phase  converter.  What  is  the 
magnetic  flux  per  field-pole  ? 

550  volts  at  11  volts  per  commutator  segment  gives  50, 
or  as  next  integer  divisible  by  3,  n  —  51  segments  or  turns 
per  pole. 

8  poles  give  4  cycles  per  revolution,  500  revolutions  per 
minute,  or  500/60  =  8.33  revolutions  per  second.  Thus 
the  frequency  is,  N=  4  X  8.33  =  33.3  cycles  per  second. 

The  induced  E.M.F.  is  e  =  550  volts,  thus  by  the  for- 
mula of  direct  current  induction, 

e  =  4  Nn  & 

or,  550  =  4  x  .333  x  51  <l> 

=  8.1  ml  per  pole. 


V 

lat  is  theiE.M.F.  i 


(3.)  What  is  the^E.M.F.  induced  in  a  single  turn  of  a  20- 


EFFECT  AND   EFFECTIVE    VALUES.  15 

polar  alternator  revolving  at    200   revolutions    per   minute, 
through  a  magnetic  field  of  6.4  ml  per  pole  ? 

20  v  200 

The  frequency  is  N=  -       ^-  =  33.3  cycles. 
Jj  X  ou 

e  =  EQ  sin  <j> 


y  -LS- 

«,,_; 

=  6.4 
r=  .333 


Thus,          £Q  —  2  TT  x  .333  x  6.4  =  13.3  volts  maximum,  or 

e  ==  13.3  sin  </>. 

4.    EFFECT   AND    EFFECTIVE   VALUES. 

The  effect  or  power  of  the  continuous  E.M.F.  E  produ- 
cing continuous  current  /  is  P  =  EL 

The  E.M.F.  consumed  by  resistance  r  is  E^  —  Ir,  thus 
the  effect  consumed  by  resistance  r  is  P  =  Pr. 

Either  E^  =  E,  then  the  total  effect  of  the  circuit  is  con- 
sumed by  the  resistance,  or  E^  <  E,  then  only  a  part  of  the 
effect  is  consumed  by  the  resistance,  the  remainder  by  some 
counter  E.M.F.  E  —  Ev 

If  an  alternating  current  z'  =  /0  sin  <f>  passes  through  a 
resistance  r,  the  effect  consumed  by  the  resistance  is, 


Pr  =  I*r  sin2  <£  =  -^-  (1-  cos 

thus  varies  with  twice  the  frequency  of  the  current,  between 
zero  and  70V. 

The  average  effect  consumed  by  resistance  r  is, 


Since  avg.  (cos)  =  0. 

Thus  the  alternating  current  /  =  70  sin  <£  consumes  in  a 
resistance  r  the  same  effect  as  a  continuous  current  of 
intensity  ^  ^ 


16  ELECTRICAL    ENGINEERING. 

The  value  I  —  -~  is  called  the    effective  value  of  the 
alternating  current  i  =  70  sin  <£,  since  it  gives  the  same  effect. 

£ 

Analogously  E  =  —  =  is  the  effective  value  of  the  alternat- 

ing E.M.F.  e  =  EQ  sin  <£. 

Since  E0  =  ^Nn^,  it  follows  that 

E= 


the  effective  alternating  E.M.F.  induced  in  a  coil  of  ;/  turns 
rotating  with  frequency  N  (in  hundreds  of  cycles)  through 
a  magnetic  field  of  $  megalines  of  force. 

This  is  \^  formula  of  alternating  current  induction. 

The  formula  of  direct  current  induction, 


holds  also  if  the  E.M.F.'s  induced  in  the  individual  turns  are 
not  sine  waves,  since  it  is  the  average  induced  E.M.F. 
The  formula  of  alternating  current  induction, 

E  =    V2irJV«$, 

does  not  hold  if  the  waves  are  not  sine  waves,  since  the 
ratios  of  average  to  maximum  and  of  maximum  to  effective 
E.M.F.  are  changed. 

If  the  variation  of  magnetic  flux  is  'not  sinusoidal,  the 
effective  induced  alternating  E.M.F.  is, 

E  =  y  ^TrN?i^. 

y  is  called  the  "form  factor"  of  the  wave,  and  depends 
upon  its  shape,  that  is  the  distribution  of  the  magnetic  flux 
in  the  magnetic  field. 

EXAMPLES. 

(1.)  In  a  star  connected  20  polar  three-phaser,  revolving 
at  33.3  cycles  or  200  revolutions  per  minute,  the  magnetic 
flux  per  pole  is  6.4  ml.  The  armature  contains  one  slot  per 
pole  and  phase,  and  each  slot  contains  36  conductors.  All 


EFFECT  AND   EFFECTIVE    VALUES.  17 

these  conductors  are  connected  in  series.  What  is  the 
effective  E.M.F.  per  circuit,  and  what  the  effective  E.M.F. 
between  the  terminals  of  the  machine  ? 

Twenty  slots  of  36  conductors  give  720  conductors,  or 
360  turns  in  series.     Thus  the  effective  E.M.F.  is, 


=  4.44  x  .333  x  360  x  6.4 
=  3400  volts  per  circuit. 

The  E.M.F.  between  the  terminals  of  a  star  connected 
three-phaser  is  the  resultant  of  the  E.M.F.'s  of  the  two 
phases,  which  differ  by  60°,  and  is  thus  2  sin  60°  =  V3  times 
that  of  one  phase,  thus, 

£  =  £l^/3 

=  5900  volts  effective. 

(2.)  The  conductor  of  the  machine  has  a  section  of  .22 
cm.2  and  a  mean  length  of  240  cm.  per  turn.  At  a  resis- 
tivity (resistance  per  unit  section  and  unit  length)  of  copper 
of  p  =  1.8  X  10~6,  what  is  the  E.M.F.  consumed  in  the 
machine  by  the  resistance,  and  what  the  energy  consumed 
at  450  K.W.  output  ? 

450  K.W.  output  is  150,000  watts  per  phase  or  circuit, 

150,000 
thus  the  current  /=     0400     =  44.2  amperes  effective. 

The  resistance  of  360  turns  of  240  cm.  length,  .22  cm.2 
section  and  1.8  X  10~6  resistivity,  is 

360  x  240  x  1.8  x  10-* 
r  =  —  —  fj~  —  -  =  .  1  1  ohms  per  circuit. 

44.2  amps.  X  .71  ohms  gives  31.5  volts  per  circuit  and  44.  22 
x  .71  =  1400  watts  per  circuit,  or  a  total  of  3  X  1400 
=  4200  watts  loss. 

(3.)  What  is  the  self-induction  per  wire  of  a  three-phase 
line  of  14  miles  length  consisting  of  three  wires  No.  0 
(d—  .82  cm.),  45  cm.  apart,  transmitting  the  output  of  this 
450  K.W.  5900  volt  three-phaser  ? 


18  ELECTRICAL   ENGINEERING. 

450  K.W.  at  5900  volts  gives  44.2  amperes  per  line. 
44.2  amperes  effective  gives  44.2  V2  =  62.5  amperes  maxi- 
mum. 

14  miles  =  22300  m.  The  magnetic  flux  produced  by 
/  amperes  in  1000  m.  of  a  transmission  line  of  2  wires  45  cm. 
apart  and  .82  cm.  diameter  was  found  in  paragraph  1,  ex- 
ample 3,  as  2  $  =  .188  x  106/,  or  $>  =  .094  x  106/for  each 
wire. 

Thus  at  22300  m  and  62.5  amperes  maximum  it  is  per 

wire, 

$  =  22.3  x  62.5  x  .094  x  106  =  131  ml. 

Hence  the  induced  E.M.F.,  effective  value,  at  33.3 
cycles  is, 


=  4.44  x  .333  x  131 
=  193  volts  per  line  ; 

the  maximum  value  is, 

EQ  =  E  X  V2  =  273  volts  per  line  ; 
and  the  instantaneous  value, 

e  =  £0  sin  (4  -  <k)  =  273  sin  (<£  -  <k); 
or,  since  <£  =  2  irNt  =  210  ^  we  have, 

e  =  273  sin  210  (/  -  /i). 

5.    SELF-INDUCTION   AND   MUTUAL   INDUCTION. 

The  number  of  inter  linkages  of  an  electric  circuit  with 
the  lines  of  magnetic  force  of  the  flux  produced  by  unit 
current  in  the  circuit  is  called  the  inductance  of  the  circuit. 

The  number  of  interlinkages  of  an  electric  circuit  with 
the  lines  of  magnetic  force  of  the  flux  produced  by  unit 
current  in  a  second  electric  circuit  is  called  the  mutual  in- 
ductance of  the  second  upon  the  first  circuit.  It  is  equal  to 
the  mutual  inductance  of  the  first  upon  the  second  circuit, 
as  will  be  seen,  and  thus  called  the  mutual  inductance  be- 
tween the  two  circuits. 


SELF-INDUCTION  AND  MUTUAL   INDUCTION.  19 

The  number  of  interlinkages  of  an  electric  circuit  with 
the  lines  of  magnetic  force  of  the  flux  produced  by  unit 
current  in  this  circuit  and  not  interlinked  with  a  second 
circuit  is  called  the  self-inductance  of  the  circuit. 

If  i  =  current  in  a  circuit  of  n  turns,  <E>  =  flux  produced 
thereby  and  interlinked  with  the  circuit,  «3>  is  the  total 

n  & 
number  of  interlinkages,  and  L  =  —  ^  the  inductance  of  the 

circuit. 

If  3>  is  proportional  to  the  current  i  and  the  number  of 
turns  «, 

2 

<£  =  —  ,  and  L  =  —  the  inductance. 
P  P 

p  is  called  the  reluctance  and  ni  the  M.M.F.  of  the  mag- 
netic circuit. 

The  reluctance  p  has  in  magnetic  circuits  the  same  posi- 
tion as  the  resistance  r  in  the  electric  circuit. 

The  reluctance  p,  and  thus  the  inductance,  is  constant 
only  in  circuits  containing  no  magnetic  materials,  as  iron, 
etc. 

If  pl  is  the  reluctance  of  a  magnetic  circuit  interlinked 
with  two  electric  circuits  of  n^  and  #2  turns  respectively,  the 
flux  produced  by  unit  current  in  the  first  circuit  and  inter- 

linked with  the  second  circuit  is  —  and  the   mutual  induc- 

Pi 

tance  of  the  first  upon  the  second  circuit,  thus,  M  =  -^—  2> 

Pi 
that  is,  equal  to  the  mutual  inductance  of  the  second  circuit 

upon  the  first  circuit,  as  stated  above. 

If  ft  =  p,  that  is,  no  flux  passes  between  the  two  circuits, 
and  L1  =  inductance  of  the  first,  L2  =  inductance  of  the 
second  circuit  and  M=  mutual  inductance,  then 


If  ft  >  p,  that  is,  if  flux  passes  between  the  two  circuits, 
then  M2  <  L^LV 

In  this  case,  the  total  flux  produced  by  the  first  circuit 


20  ELECTRICAL   ENGINEERING 

consists  of  a  part  interlinked  with  the  second  circuit  also, 
the  mutual  inductance,  and  a  part  passing  between  the  two 
circuits,  that  is,  interlinked  with  the  first  circuit  only,  its 
self  -inductance. 

Thus,  if  Ll  and  Lz  are  the  inductances  of  the  two  cir- 

cuits, —  -  and  —  -  is  the  total  flux  produced  by  unit  current 
«!  nz 

in  the  first  and  second  circuit  respectively. 

L  S 

Of  the  flux  --  a  part  —  is  interlinked  with  the  first  cir- 
«i  *i 

M 
cuit  only,  and  Sl  called  its  self-inductance,  and  the  part  - 

nz 

interlinked  with  the  second  circuit  also,  where  M  =  mutual 

,  A       S.      M 

inductance,  and  —  =  —  +  — 
»i       «i       ». 
Thus,  if, 

Zj  and  Z2  =  inductance, 
S-i  and  S2  =  self-inductance, 

J/=  mutual  inductance  of  two  circuits  of  n±  and 
n2  turns  respectively,  we  have, 


or,  L^S.  +      M  LZ  =  S2  +      M 

HI  n^ 

or,  J/2  =  (A  -  S,)  (Z2  -  $).  ' 

The  practical  unit  of  inductance  is  109  times  the  absolute 
unit  or  108  times  the  number  of  interlinkages  per  ampere 
(since  1  amp.  =  .1  unit  current),  and  is  called  the  henry  ; 
.001  of  it  is  called  the  milhenry  (mh). 

The  number  of  interlinkages  of  i  amperes  in  a  circuit  of 
L  henry  inductance  is  iL  108  lines  of  force  turns,  and  thus 
the  E.M.F.  induced  by  a  change  of  current  di  in  time  dt  is 

e=  —  —  L  108  absolute  units 
dt 

=  -*Z  volts. 


SELF-INDUCTION  AND   MUTUAL   INDUCTION.          21 

A  change  of  current  of  one  ampere  per  second  in  the 
circuit  of  one  henry  inductance  induces  one  volt. 


EXAMPLES. 

(1.)  What  is  the  inductance  of  the  field  of  a  20  polar 
alternator,  if  the  20  field  spools  are  connected  in  series, 
each  spool  contains  616  turns,  and  6.95  amperes  produces 
6.4  ml.  per  pole  ? 

The  total  number  of  turns  of  all  20  spools  is  20  X  616 
=  12320.  Each  is  interlinked  with  6.4  X  106  lines,  thus 
the  total  number  of  interlinkages  at  6.95  amperes  is  12320 
x  6.4  x  106  =  78  x  109. 

6.95  amperes  =  .695  absolute  units,  hence  the  number 
of  interlinkages  per  unit  current,  or  the  inductance,  is, 

78  x  10' 


.695 


=  112  x  109  =  112  henrys. 


(2.)  What  is  the  mutual  inductance  between  an  alternat- 
ing transmission  line  and  a  telephone  wire  carried  for  10 
miles  below  and  1.20  m.  distant  from  the  one,  1.50  m.  dis- 
tant from  the  other  conductor  of  the  alternating  line  ?  and 
what  is  the  E.M.F.  induced  in  the  telephone  wire,  if  the 
alternating  circuit  carries  100  amperes  at  60  cycles  ? 

The  mutual  inductance  between  the  telephone  wire  and 
the  magnetic  circuit  is  the  magnetic  flux  produced  by  unit 
current  in  the  telephone  wire  and  interlinked  with  the  alter- 
nating circuit,  that  is,  that  part  of  the  magnetic  flux  produced 
by  unit  current  in  the  telephone  wire,  which  passes  between 
the  distances  of  1.20  and  1.50  m. 

At  the  distance  x  from  the  telephone  wire  the  length  of 

magnetic  circuit  is  ^-KX.     The  magnetizing  force /=  ~ — 

if  /  =  current  in  telephone  wire  in  amperes,  and  the  field 

.27 
intensity  3C  =  Airf  = ,  and  the  flux  in  the  zone  dx, 


22  ELECTRICAL   ENGINEERING. 

2  II 

d$>  =  ^  ±  dx 

x 

I  =  10  miles  =  1610  x  108  cm. 


f150.2/ 
=  /  - 

t/    120        X 


X 

=  322  x  103/log6         =  72  7103 


thus, 


or,          72  /103  interlinkages,  hence,  for  /=  10,  or  one  absolute^ 

unit, 
thus,     M  =  72  x  104   absolute   units,  =  72  x  10~5   henrys  = 

.72  mh. 

100  amperes  effective  or  141.4  amperes  maximum  or 
14.14  absolute  units  of  current  in  the  transmission  line  pro- 
duces a  maximum  flux  interlinked  with  the  telephone  line  of 
14.14  x  .72  x  10-8  x  109  =  10.2  ml. 

Thus  the  E.M.F.  induced  at  60  cycles  is, 

E  =  4.44  x  .6  x  10.2  =  27.3  volts  effective. 


6.    SELF-INDUCTION   OF    CONTINUOUS    CURRENT 
CIRCUITS. 

Self-induction  makes  itself  felt  in  continuous  current 
circuits  only  in  starting  and  stopping  or  in  general  changing, 
the  current. 

Starting  of  Current  : 

If    r  =  resistance, 

L  =  inductance  of  circuit, 

E  =  continuous  E.M.F.  impressed  upon  circuit, 
i  —  current    in    circuit    at    time    /    after    impressing 
E.M.F.  E, 

and  di  the  increase  of  current  during  time  moment  dt,  theni 
Increase  of  magnetic  interlinkages  during  time  dty 

Ldi. 
E.M.F.  induced  thereby, 

di 


SELF-INDUCTION  OF  CONTINUOUS   CIRCUITS.  23 

negative,  since  opposite  to  the  impressed  E.M.F.  as  its  cause, 
by  Lentz's  law. 

Thus  the  E.M.F.  acting  in  this  moment  upon  the  circuit 

is, 

E  +  ei  =  M-Ljt, 

and  the  current, 

E-L- 

.  _  E  -f  el  _  dt 

r  r 

or  transposed, 

rdt  _      di 

~T~      ~' 

t 

r 

The  integral  of  which  is, 

rt  I .      E\      . 

~  z  =    ge  \  ~  7/  ~    ge  fm 

Where  —  loge  c  =  integration  constant. 
This  reduces  to, 

•      £  ,      -- 

t  = h  C*.      L 

r 
at  t  =  0,  i  =  0,  and  thus, 


Substituting  this  value, 

J?  i  r  t\ 

i  =  —  f  1  —  e~^J  the  current, 

_jri 

e1  =  ir  —  £=— £e    Z  the  E.M.F.  of  self-induction. 
At  /  =  oo  , 

4  =  —,         e1  =  Q. 

Substituting  these  values, 

/          —rJ.\ 
i  =  /0  U  —  c     1) 

rt 


The  expression    T0  =  —  is  called  the  "  //w^  constant  of 
the  circuit." 


24  ELECTRICAL   ENGINEERING. 


Substituted,  in  the  foregoing  equation,  this  gives, 

-.-*) 


At  t  =  T0, 


-!. 


el  =  -  —  =  -  .368  E. 


Stopping  of  Current  : 

In  a  circuit  of  inductance  L  and  resistance  r,  let  a  cur- 

j^ 
rent  z'0  =  -:  be  produced  by  the  impressed  E.M.F.  E,  and 

this  E.M.F.  E  be  withdrawn  and  the  circuit  closed  by  a 
resistance  rv 

Let  the  current  be  /.at  the  time  t  after  withdrawal  of  the 
E.M.F.  E  and  the  change  of  current  during  time  moment 
dt  be  di. 

di  is  negative,  that  is,  the  current  decreases. 

The  decrease  of  magnetic  interlinkages  during  moment 

dt\$, 

Ldi. 

Thus  the  E.M.F.  induced  thereby, 

di 
e^~L^t 

negative,  since  di  is  negative  and  ^  must  be  positive  or  in 
the  same  direction  as  E,  to  maintain  the  current  or  oppose 
the  decrease  of  current  as  its  cause. 
The  current  is  then, 

.         e\  L     di 

r  +  *i  r  -f-  rl  dt 

or  transposed, 


L  z 

the  integral  of  which  is 

—  r  ^l  t  =  loge  /  —  loge< 
where  —  loge  c  =  integration  constant. 


SELF-INDUCTION  OF  CONTINUOUS   CIRCUITS.          25 

This  reduces  to        /"=  <re       IT* 

for,  /  =  o, 

r 

Substituting  this  value,  we  have, 

E     _(r±r£t 

i  =  —  e        L       the  current, 

'i  =  i(r  +  ^i)  =  ^^t^'c^^HH  the  induced  E.M.F. 
Substituting  /<,  =  —,  we  have, 

*'=  /ie       Z~^  the  current, 

e\  =  ?o  (r  +  ri)  e       ^^  the  induced  E.M.F. 

At  /  =  0, 


That  is,  the  induced  E.M.F.  is  increased  over  the  pre- 
viously impressed  E.M.F.  in  the  same  ratio  as  the  resistance 
is  increased. 

When  ^  =  0,  that  is,  when  in  withdrawing  the  impressed 
E.M.F.  E  the  circuit  is  short-circuited, 

E  _rJ.  _^ 

t  =  —  €    L  =  /Q€    L  the  current,  and 

—  —  —r-l 

el  =  Ef.     L  =  iQre     L  the  induced  E.M.F. 


In  this  case,  at  t  =  0,  el  =  E,  that  is,  the  E.M.F.  does  not 
rise. 

In  the  case,  r  =  oo  ,  that  is,  if  in  withdrawing  the  E.M.F. 
E  the  circuit  is  broken,  we  have, 

at  /  =  0,  el  =  oo  ,  that  is,  the  E.M.F.  rises  infinitely. 

The  greater  rv  the  higher  is  the  induced  E.M.E.  ev  the 
faster,  however,  e  and  i  decrease. 
If  r^  —  r,  we  have  at  t  =  0, 

eu  =  2£,  i—  t'0, 

and, 


26  ELECTRICAL   ENGINEERING. 

that  is,  if  the  external  resistance  ^  equals  the  internal  resist- 
ance r,  in  the  moment  of  withdrawal  of  E.M.F.  E  the  ter- 
minal voltage  is  E. 

The  effect  of  the  E.M.F.   of   self-induction  in  stopping 
the  current  is  at  the  time  /, 

^i=/o2(r+ri)e-^' 

thus  the  total  energy  of  the  induced  E.M.F. 
W=         ie^dt 


that  is,  the  energy  stored  as  magnetism  in  a  circuit  of  cur- 
rent i0  and  self  -inductance  L,  is, 


which  is  independent  both  of  the  resistance  r  of  the  circuit, 
and  the  resistance  r^  inserted  in  breaking  fhe  circuit.  This 
energy  has  to  be  expended  in  stopping  the  current. 

EXAMPLES. 

(1.)  In  the  alternator  field  in  section  1,  example  4,  sec- 
tion 2,  example  2,  and  section  5,  example  1,  how  long  time 
after  impressing  the  required  E.M.F.  E  =  230  volts  will  it 
take  for  the  field  to  reach, 

0.)    -j-  strength, 
(J.)    ^strength? 

(2.)  If  500  volts  are  impressed  upon  the  field  of  this 
alternator,  and  a  noninductive  resistance  inserted  in  series 
so  as  to  give  the  required  exciting  current  of  6.95  amperes, 
how  long  after  impressing  the  E.M.F.  E  =  500  volts  will  it 
take  for  the  field  to  reach, 

(a.)   i  strength, 

(J.)    A  strength, 

(f.)    and  what  Is  the  resistance  required  in  the  rheostat  ? 


SELP-INDUCTION  OF  CONTINUOUS   CIRCUITS.          27 

(3.)  If  500  volts  are  impressed  upon  the  field  of  this 
alternator  without  insertion  of  resistance,  how  long  will  it 
take  for  the  field  to  reach  full  strength  ? 

(4.)  With  full  field  strength  what  is  the  energy  stored 
as  magnetism  ? 

(1.)  The  resistance  of  the  alternator  field  is  33.2  ohms 
(section  2,  example  2),  the  inductance  112  h.  (section  5, 
example  1),  the  impressed  E.M.F.  is  E  =  230,  the  final 

value  of  current  z'0  =  —  =  6.95  amperes.     Thus  the  current 
at  time  /, 

=  6.95  (1  -  e— **<). 
a.)  £  strength,  *  =  | ,  hence  (1  -  e--296<)  =  .5 

e-2881  =  .5,  -  .296  /  =  Iog6  .5  =  -  .693 
after  /  =  .234  seconds. 

b.)   T%    strength:    /  =  .9  iot    hence    (1  -  e-296')  =  .9,    after 
=  7.8  seconds. 

(2.)    To  get  *0  =  6.95  amperes,  with  E  =  500  volts,  a 

resistance  r=  ^  =  72  ohms,  and  thus  a  rheostat  of  72 

-  33.2  =  38.8  ohms  is  required. 
We  then  have, 

.__    .    /^  _       _rt\ 

=  6.95  (1  -  e-643). 
a.)   *  =  | »  after  /  =  .108  seconds. 
b.)    i  —  .9  /Q,  after  /  =  .36  seconds. 

(3.)  Impressing  E  =  500  volts  upon  a  circuit  of  r  =  33.2, 
L  =  112  gives, 


=  15.1  (1  -  c-296') 
/  =  6.95,  or  full  field  strength,  gives, 
6.95  =  15.1  (1  -  €--296') 
1  -  e-298'  =  .46 

after  /  =  2.08  seconds. 


28  ELECTRICAL   ENGINEERING. 

(4.)    The  stored  energy  is, 

/02Z      6.952  x  112 

-o-  :          —  n  —    -  =  ^720  watt  seconds  or  joules, 

=  2000  foot  pounds. 

Thus  in  case  (3),  where  the  field  reaches  full  strength  in 

2000 

2.08  seconds,  the  average  power  input  is  -  =  960  foot 

2.08 

pound  seconds,  =  If  H.P. 

In  breaking  the  field  circuit  of  this  alternator,  2000  foot 
pounds  have  to  be  destroyed,  in  the  spark,  etc. 

(5.)  A  coil  of  resistance,  r  =  .002  ohms,  and  inductance 
L  =  .005  milhehrys,  carrying  current  /=  90  amperes,  is 
short-circuited. 

a.)    What  is  the  equation  of  the  current  after  short-circuit  ? 
If.)    In  what  time  has  the  current  decreased  to  y1^  its  initial 
value  ? 

_  ri 

a.)    i  =  If.      L 

=  90  e-400' 
b.)    i  =  .1  7,  e-400'  =  .1,  after  /  =  .00576  seconds. 

(6.)  When  short-circuiting  the  coil  in  example  5,  an 
E.M.F.  E  =  1  volt  is  inserted  in  the  circuit  of  this  coil,  in 
opposite  direction  to  the  current. 

a.)   What  is  equation  of  the  current  ? 

£.)    After  what  time  has  the  current  become  zero  ? 

e.)  After  what  time  has  the  current  reverted  to  its  initial 
value  in  opposite  direction  ? 

//.)  What  impressed  E.M.F.  is  required  to  make  the  current 
die  out  in  -5^-$  second  ? 

e.)  What  impressed  E.M.F.  E  is  required  to  reverse  the  cur- 
rent in  second  ? 


a.)  If  E.M.F.  —  E  is  inserted,  and  at  time  /  the  current 
is  denoted  by  it  we  have, 

el  =  —  L  -!r  the  induced  E.M.F. 


SELF-INDUCTION  IN  ALTERNATING    CIRCUITS.        29 

thus, 

-  E  +  el  =  -  E  —  L  j  the  total  E.M.F. 
and, 


transposed, 


/  =  -  —  — -r-  the  current 

r  r        r  dt 


integrated, 


where  —  log,  c  —  integration  constant. 

At  /  =  0 
substituted, 


At  /  =  0,  i  =  /,  thus  c  =  I  +  — 

r 


_-'_*      E 


i  =  590  e-400'  -  500. 

£.)  ,'=  0,  e-*»'=  .85,  after  /  =  .000405  seconds.  . 
c.)   i=  -  1=  -  90,  e-400'  =  .694,  after  /=  .00091  seconds. 
</.)  if    z  =  0  at  /  =.0005,  then 

0  =  (90  +  500  £)  e-  2  -  500  £, 

E  =  ^  2^     =  .81  volts. 

<?.)  if     /  =  -  7=  -  90  at  /  =  .001,  then, 

-  90  =  (90  +  500  JE)  c--4  -  500  E, 


7.    SELF-INDUCTION    IN    ALTERNATING    CURRENT 

CIRCUITS. 

An  alternating  current  i  =  70  sin  2  TT Nt  or  z  =  70  sin  <#> 
can  be  represented  graphically  in  rectangular  coordinates  by  a 
curve  line  as  shown  in  Fig.  6,  with  the  instantaneous  values 
i  as  ordinates  and  the  time  /,  or  the  arc  of  the  angle  corre- 


30 


ELECTRICAL   ENGINEERING. 


spending  to  the  time,  <f>  =  2  vNt  as  abscissae,  counting  the 
time  from  the  zero  value  of  the  rising  wave  as  zero  point. 


3_T 


Fig.  6. 


If  the  zero  value  of  current  is  not  chosen  a"s  zero  point 
of  time,  the  wave  is  represented  by, 


or, 


=      sn    <   - 


where  t  =  /0,  or  <£  =  <f>0  is  the  time  and  the  corresponding 
angle,  at  which  the  current  reaches  its  zero  value  in  the 
ascendant. 

If  such  a  sine  wave  of  alternating  current  i  =  70  sin 
2  irNt  or  i  =  70  sin  <£,  passes  through  a  circuit  of  resistance 
r  and  inductance  L,  the  magnetic  flux  produced  by  the  cur- 
rent and  thus  its  interlinkages  with  the  current,  iL  =  IQL 


sin  <£,  vary  in  a  wave  line  similar  also  to  that  of  the  current, 
as  shown  in  Fig,  7  as  3>.  The  E.M.F.  induced  hereby  is  pro- 
portional to  the  change  of  iL,  and  is  thus  a  maximum  where 
iL  changes  most  rapidly,  or  at  its  zero  point,  and  zero  where 


SELF-INDUCTION  IN  ALTERNATING   CIRCUITS.        31 

iL  is  a  maximum.  It  is  positive  during  falling,  negative 
during  rising  current,  by  Lentz's  Law.  Thus  this  induced 
E.M.F.  is  a  wave  following  the  wave  of  current  by  the  time 

/  =  _    where  T  is  time  of  one  complete  period,  =  -^,  or  by 

the  angle  <t>  =  90°. 

This  E.M.F.  is  called  the  counter  E.M.F.  of  self-induc- 
tion.    It  is, 


cos 

It  is  shown  in  dotted  line  in  Fig.  7  as  e£. 

The  quantity,  2  irNL,  is  called  the  reactance  of  the  cir- 
cuit, and  denoted  by  x.  It  is  of  the  nature  of  a  resistance, 
and  expressed  in  ohms. 

If  L  is  given  in  109  absolute  units  or  henrys,  x  appears 
in  ohms. 

The  counter  E.M.F.  of  self-induction  of  current 

/  =  JQ  sin  2  TtNt  =  IQ  sin  <£ 
of  effective  value, 

/    7» 
1  vr 

is, 

<?2'  =  —  ^  cos  2  TT^W  =  —  x!Q  cos  <£ 

of  maximum  value, 

xiQ 

and  effective  value, 

T?  X*0  T 

£2=  -  p  =  ^/. 

V2 

That  is,  the  effective  value  of  the  counter  E.M.F.  of 
self-inductance  equals  the  reactance  x  times  the  effective 
value  of  the  current  7,  and  is  lagging  90°  or  a  quarter  period 
behind  the  current. 

By  the  counter  E.M.F.  of  self-induction, 

e2'  =  —  xIQ  cos  <£ 


32  ELECTRICAL   ENGINEERING. 

which  is  induced  by  the  passage  of  the  current  i  =  70  sin  <£ 
through  the  circuit  of  reactance  x,  an  equal  but  opposite 

E.M.F. 

<?2  =  xSQ  cos  <£ 

is  consumed,  and  thus  has  to  be  impressed  upon  the  circuit. 
This  E.M.F.  is  called  the  E.M.F.  consumed  by  self  induction. 
It  is  90°  or  a  quarter  period  ahead  of  the  current,  and  shown 
in  Fig.  1  as  drawn  line  ev 

Thus  we  have  to  distinguish  between  counter  E.M.F.  of 
self-induction,  90°  lagging,  and  E.M.F.  consumed  by  self- 
induction,  90°  leading. 

These  E.M.F.'s  stand  in  the  same  relation  as  action  and 
reaction  in  mechanics.  They  are  shown  in  Fig.  1  as  ez  and 
as  e£. 

The  E.M.F.  consumed  by  the  resistance  r  of  the  circuit 
is  proportional  to  the  current, 

e1  =  ri  =  r!Q  sin  <£, 

and  in  phase  therewith,  that  is,  reaches  its  maximum  and  its 
zero  value  at  the  same  time  as  the  current  z,  as  shown  by 
drawn  line  ^  in  Fig.  1. 

Its  effective  value  is  E^  =  ri. 

The  resistance  can  also  be  represented  by  a  (fictitious) 
counter  E.M.F., 

<?/  =  —  rf0  sin  <£ 

opposite  in  phase  to  the  current,  shown  as  ^/  in  dotted 
line  in  Fig.  4. 

The  counter  E.M.F.  of  resistance  and  the  E.M.F.  con- 
sumed by  resistance  have  the  same  relation  to  each  other  as 
the  counter  E.M.F.  of  self. -induct ion  and  the  E.M.F.  con- 
sumed by  self-induction  or  reactance. 

If  an  alternating  current  i  =  70  sin  <£  of  effective  value 

/  =  — =  passes  through  a  circuit  of  resistance  r  and  induc- 
tance Z,  that  is,  reactance  x  =  2-n-NL,  we  have  thus  to  dis- 
tinguish : 


SELF-INDUCTION  IN  ALTERNATING    CIRCUITS.        33 

E.M.F.  consumed  by  resistance,  el  =  rIQ  sin  <f>,  of  effect- 
ive value  E^  =  rl,  and  in  phase  with  the  current. 

Counter  E.M.F.  of  resistance,  <?/  =  —  rIQ  sin  <£,  of 
effective  value  E^  =  rf,  and  in  opposition  or  180°  displaced 
from  the  current. 

E.M.F.  consumed  by  reactance,  e2  =  xIQ  cos  <j>,  of  effect- 
ive value  Ez  =  xl,  and  leading  the  current  by  90°  or  a 
quarter  period. 

Counter  E.M.F.  of  reactance,  ea'  =  —  xIQ  cos  <£,  of  effect- 
ive value  E±  =  xly  and  lagging  90°  or  a  quarter  period 
behind  the  current. 

The  E.M.F.'s  consumed  by  resistance  and  by  reactance 
are  the  E.M.F.'s  which  have  to  be  impressed  upon  the  cir- 
cuit to  overcome  the  counter  E.M.F.'s  of  resistance  and  of 
reactance. 

The  total  counter  E.M.F.  of  the  circuit  is  thus, 

/  =  e{  +  <?/  =  —  f0  (r  sin  <f>  +  x  cos  <j>) 

and  the  total  impressed  E.M.F.,  or  E.M.F.  consumed  by 

the  circuit, 

e  =  ^  +  ez  =  IQ  (r  sin  <f>  -f  x  cos  <#>). 
Substituting 

* 

-  =  tan  o>,  and 

Vr*  +  x9  =  z, 
from  which  follow 

x  =  z  sin  a),  r  =  z  cos  CD, 

we  have, 
total  impressed  E.M.F. 

e  =  zf0  sin  (<£  -j-  <o), 

shown  by  heavy  drawn  line  e  in  Fig.  7,  and  total  counter 

E.M.F. 

/  =  —  zIQ  sin  (<£  +  o>), 

shown  by  heavy  dotted  line  e1  in  Fig.  7,  both  of  effective 
value, 


34:  ELECTRICAL   ENGINEERING. 

For  <£  =  —  <D,  e  =  0,  that  is,  the  zero  value  of  e  is  by 
angle  w  ahead  of  the  zero  value  of  current,  or  the  current 
lags  behind  the  impressed  E.M.F  by  angle  CD. 

o>  is  called  the  angle  of  lag  of  the  current,  and  s 
—  V;2  +  x*  the  impedance  of  the  circuit,  e  is  called  the 
E.M.F.  consumed  by  impedance,  £  the  counter  E.M.F.  of 
impedance. 

Since     E^  =  r/is  the  E.M.F.  consumed  by  resistance, 
Ez  —  ^t'/is  the  E.M.F.  consumed  by  reactance, 
and  E  =  zl=  Vr2  H-  oP  I  is  the  E.M.F.  consumed  by 

impedance, 
we  have, 


E  =      ^2  +  E?,  the  total  E.M.F. 
and  El  —  E  cos  <o 

Ez  =  E  sin  to  its  components. 

The  tangent  of  the  angle  of  lag  is, 


x 

tan  o>  =  -  = 
r 


and  the  time  constant  of  the  circuit  is, 

L      tan  <o 


The  total  E.M.F.  impressed  upon  the  circuit,  e,  consists 
of  two  components,  one,  ev  in  phase  with  the  current,  the 
other  one,  ey  in  quadrature  with  the  current. 

Their  effective  values  are, 

E,  E  cos  (o,  E  sin  CD. 

EXAMPLES. 

(1.)  What  is  the  reactance  per  wire  of  a  transmission  line 
of  length  /,  if  d  =  diameter  of  the  wire,  D  =  distance  be- 
tween wires,  and  N  =  frequency  ? 

If  /  =  current,  in  absolute  units,  in -one  wire  of  the  trans- 
mission line,  the  M.M.F.  is  /,  thus  the  magnetizing  force  in  a 
zone  d*  at  distance  x  from  center  of  wire  (Fig.  8)  is  f  = 


SELF-INDUCTION  IN  ALTERNATING*' CIRCUITS.         35 


s  —  ,  and  the  field  intensity  in  this  zone  is  3C  ==  4?r/=  2Z. 

2  7TX  X 

Thus  the  magnetic  flux  in  this  zone, 


hence  the  total   magnetic   flux  between   the  wire  and  the 
return  wire, 

/n  CD  d* 

,'*-™f  ,"-*"* 

2  2 

neglecting  the  flux  inside  the  transmission  wire. 


Fig.    8. 


The  coefficient  of  self-induction  or  inductance  is  thus, 
L  =  —  =  2  I  log€  —  —  absolute  units, 


=  2/loge 


and  the  reactance  x=  2-irN'L  =  47rJWloge  —  —  absolute  units 

2  D 

=  4  VJW  log,  --  10  ~9  ohms. 


(2.)    The  voltage  at  the  receiving  end  of  a  33.3  cycle 
transmission  of  14  miles'  length  shall  be  5500  between  the 


36  ELECTRICAL   ENGINEERING. 

lines.  The  transmission  consists  of  three  wires,  (No.  0 
B.  &  5.  G.)  (d=  .82  cm),  18"  (45  cm.)  apart,  of  resistivity  P 
=  1.8  x  10-6. 

a.)  What  is  the  resistance,  the  reactance,  and  the  impedance 
per  line,  and  the  voltage  consumed  thereby  at  44  amperes  flow- 
ing over  the  line  ? 

b.)  What  is  the  generator  voltage  between  lines  at  44  am- 
peres flowing  into  a  non-inductive  circuit  ? 

f.)  What  is  the  generator  voltage  between  lines  at  44  am- 
peres flowing  into  a  circuit  of  45°  lag  ? 

d.)  What  is  the  generator  voltage  between  lines  at  44  am- 
peres flowing  into  a  circuit  of  45°  lead  ? 

Here,  /  =  14  miles  =  14  x  1.6  X  1C5  =  2.23  x  1C6  cm. 

d  =  .82  cm. 
Hence  section    s  —  .528  cm2. 


jy=  33.3  thus, 

N    ^     .  r  /       1.8  x  10-6  x  2.23  x  10* 

a.)    Resistance  per  line,   r=p-  =  -     -  TTKOO~~ 

S  O.oZo 

=  7.60  ohms. 

2  D 
Reactance  per  line,  x  =  4kyNl\o&—j-  X  10~9=  4?r  x  3.33 

X  2.23  x  106  x  loge  110  x  10-9  =  4.35  ohms. 

The  impedance  per  line  z  =  Vr2  -f-  x?  =  8.76  ohms.    Thus 
if  /  =  44  amperes  per  line, 

E.M.F.  consumed  by  resistance,  JE1  =  rf=  334  volts. 
E.M.F.  consumed  by  reactance,  £2  =  xf=  192  volts. 
E.M.F.  consumed  by  impedance,  £s  =  zf=  385  volts. 

b.)  5500   volts   between   lines  at  receiving  circuit   give 

5500 

—  -:=-  =  3170  volts  between  line  and  neutral  or  zero  point 

(Fig.  9),  or  per  line,  corresponding  to  a  maximum  voltage 
of  3170  V2  =  4500  volts.     44  amperes  effective  per  line 
gives  a  maximum  value  of  44  V2  =  62  amperes. 
Denoting  the  current  by, 

/  =  62  sin  < 


SELF-INDUCTION  IN  ALTERNATING   CIRCUITS.        37 


the  voltage  per  line  at  the  receiving  end  with  non-inductive 
load  is  e  =  4500  sin  </>. 

The  E.M.F.  consumed  by  re- 
sistance, in  phase  with  the  current, 
of  effective  value  334,  thus  maxi- 
mum value  334  V2  =  472,  is, 

<?x  =  472  sin  <£. 

The  E.M.F.  consumed  by  re- 
actance, 90°  ahead  of  the  cur- 
rent, of  effective  value  192,  thus 
maximum  value  192  V2  =  272,  is, 

ez  =  272  cos  <f>. 

Thus  the  total  voltage  required  at  the  generator  end  of 
the  line  is,  per  line, 

e0  =  e  +  e1  +  ez=  (4500  +  472)  sin  <£  +  272  cos  <f> 
=  4972  sin  <£  +  272  cos  <£ 

272 
denoting 


Fig.    9. 


JQhJO    Lclll 

p,  we  nave, 
tan  £ 

272 

cos  ft  =  - 

v/1  +  tarT27? 
1 

4980' 
4972 

\/l  -1-  tan2  R 

4980" 

Hence, 


<?0  =  4980  (sin  <j>  cos  ft  +  cos  <f>  sin 
=  4980  sin  (d>  +  p\ 


Thus  ft  is  the  lag  of  the  current  behind  the  E.M.F.  at 
the  generator  end  of  the  line,  =  3.2°,  and  4980  the  maximum 

voltage   per   line   at   the   generator   end,    thus   EQ  =  — -=- 

v  2 

=  3520,  the  effective  voltage  per  line,  and  3520  V3=  6100 
the  effective  voltage  between  the  lines  at  the  generator. 
c.)  If  the  current, 

i  =  62  sin  <j> 

lags  45°  behind  the  E.M.F.  at  the  receiving  end  of  the  line, 
this  E.M.F.  is  expressed  by 


38  ELECTRICAL   ENGINEERING. 

e  =  4500  sin  (<£  +  45)  =  3170  (sin  <£  +  cos  <£), 

that  is,  it  leads  the  current  by  45°,  or  is  zero  at  <£  =  —  45°. 
The  E.M.F.  consumed  by  resistance  and  by  reactance 
being  the  same  as  in  b),  the  generator  voltage  per  line  is, 

eQ  =  e  +  el  +  et  =  3642  sin  </>  +  3442  cos  $ 

3442 
denoting  0        =  tan  &  we  nave> 


eQ  =  5011  sin  (<£  +  (3). 

Thus  ft  the  angle  of  lag  of  the  current  behind  the  gen- 
erator E.M.F.,  is  43°,  and  5011  the  maximum  voltage,  hence 
3550  the  effective  voltage  per  line,  and  3550  V3  =  6160 
the  effective  voltage  between  lines  at  the  generator. 

d.)  If  the  current 

i  =  62  sin  <f> 
leads  the  E.M.F.  by  45°,  the  E.M.F.  at  the  receiving  end  is, 

e  =  4500  sin  (<£  -  45) 
=  3170  (sin  cf>  —  cos  <£).  « 

Thus  at  the  generator  end 

eQ  =  e  +  ei  +  ez  =  3642  sin  <£  -  2898  cos  <£ 

2898 
denoting     /»       =  tan  p,  it  is 


e,  =  4654  sin  (<j>  -  ft). 

Thus  ft  the  angle  of  lead  at  the  generator,  =  39°,  and 
4654  the  maximum  voltage,  hence  3290  the  effective  voltage 
per  line  and  5710  the  effective  voltage  between  lines  at  the 
generator. 

8.    EFFECT   OF    ALTERNATING    CURRENTS. 
The  effect   or  power   consumed   by  alternating   current 
i  —  /o  sin  <f>,  of  effective  value  7  =  —j=  ,  in  a  circuit  of  re- 
sistance r  and  reactance  x  =  2  -n-NL,  is, 

/-.*; 


EFFECT  OF  ALTERNATING   CURRENTS.  39 

where  e  =  zIQ  sin  (<£  -f  <o)  is  the  impressed  E.M.F.  consisting 
of  the  components, 

el  =  rfQ  sin  <£  the  E.M.F.  consumed  by  resistance, 
and         <?2  =  x!Q  cos  <£  the  E.M.F.  consumed  by  reactance, 

_  j^ 

and          z  =  V/42  H-  0?  is  the  impedance,  tan    o>  =  -  the   phase 
angle  of  the  circuit. 

The  effect  is  thus, 


p  =  Z!Q  sin  <£  sin  (<£  -f-  <o) 
-.71 

=  -^  (cos  co  -  cos  (2  <£  +  o>)) 

Zi 

=  2/2  (cos  to  —  cos  (2  <£  -|-  CD)). 

Since   the   average   cos    (2  <f>  +  o>)  =  zero,    the   average 
effect  is 

/^=  2/2COS  o> 


That  is,  the  effect  of  the  circuit  is  that  consumed  by  the 
resistance,  and  independent  of  the  reactance. 

Reactance  or  self-induction  consumes  no  effect,  and  the 
E.M.F.  of  self-induction  is  a  wattless  E.M.F.,  while  the 
E.M.F.  of  resistance  is  an  energy  E.M.F. 

The  wattless  E.M.F.  is  in  quadrature,  the  energy  E.M.F. 
in  phase  with  the  current. 

In  general,  if  <o  =  phase  angle  of  circuit,  /  =  current, 
E  =  impressed  E.M.F.,  consisting  of  two  components,  one 
EI  —  E  cos  w,  in  phase  with  the  current,  the  other,  £2  =  E 
sin  o>,  in  quadrature  with  the  current,  the  effect  of  the  cir- 
cuit is  IE^  =  IE  cos  o>,  and  the  E.M.F.  in  phase  with  the 
current  E^  —  E  cos  <o  is  an  energy  E.M.F.,  the  E.M.F.  in 
quadrature  with  the  current  E2  =  E  sin  w,  a  wattless  E.M.F. 

Thus  we  have  to  distinguish  energy  E.M.F.  and  wattless 
E.M.F.,  or  energy  component  of  E.M.F.,  in  phase  with  the 
current,  and  wattless  component  of  E.M.F.,  in  quadrature 
with  the  current. 

Any  E.M.F.  can  be  considered  as  consisting  of  two  com- 
ponents, one  in  phase  with  the  current  or  energy  E.M.F.  ev 


40  ELECTRICAL   ENGINEERING. 

and  one  in  quadrature  with  the  current  or  wattless  E.M.F. 
ev     The  sum  of  instantaneous  values  of  the  two  components 

is  the  total  E.M.F. 

e  =  e1  +  e.2. 

If  E,  Ev  Ez  are  the  respective  effective  values,  we  have, 


E  =     ^1/  +  Aj2,  since 
El  =  E  cos  o> 
Ez  =  E  sin  o> 

where  w  =  phase  angle  between  current  and  E.M.F. 

Analogously,  a  current  /  passing  through  a  circuit  of 
impressed  E.M.F.  E  with  phase  angle  o>  can  be  considered 
as  consisting  of  two  component  currents, 

^  =  /  cos  a),  the  energy  current  or  energy  component  of  cur- 

rent, and, 
f2  =  /  sin  to,  the  wattless  current  or  wattless  component  of 

current. 

The  sum  of  instantaneous  values  of  energy  and  of  watt- 
less currents  equals  the  instantaneous  value  of  total  current, 

h  +  4  =  * 
while  their  effective  values  have  the  relation, 


Thus  an  alternating  current  can  be  resolved  in  two  com- 
ponents, energy  current,  in  phase,  and  wattless  current,  in 
quadrature,  with  the  E.M.F. 

An  alternating  E.M.F.  can  be  resolved  in  two  compo- 
nents, energy  E.M.F.,  in  phase,  and  wattless  E.M.F.,  in 
quadrature,  with  the  current. 

The  effect  of  the  circuit  is  the  current  times  the  E.M.F. 
times  the  cosine  of  the  phase  angle,  or  is  the  energy  current 
times  the  total  E.M.F.,  or  the  energy  E.M.F.  times  the 
total  current. 


POLAR    CO-ORDINATES.  41 

EXAMPLES. 

(1.)  What  is  the  effect  received  over  the  transmission  line 
in  section  7,  example  2,  the  effect  lost  in  the  line,  the  effect 
put  into  the  line,  and  the  efficiency  of  transmission  with  non- 
inductive  load,  45°  lag  and  45°  lead  ?. 

The  effect  received  per  line  at  noninductive  load  is  P 
=  EI=  3170  x  44  -  137  K.W. 

On  load  of  45°  phase  displacement,  P  =  El  cos  45° 
=  97  K.W. 

The  effect  lost  per  line  P1  =  PR  =  442  x  7.6  =  14.7 
K.W. 

Thus  the  input  into  the  line  PQ  =  P  +  Pl  =  151.7  K.W. 
at  noninductive  load, 

and          =  111.7  K.W.  at  load  of  45°  phase  displacement. 
The  efficiency  is  with  noninductive  load, 


With  load  of  45°  phase  displacement, 


The  total  output  is  3  P  =  411  K.W.  and  291  K.W., 
respectively. 

The  total  input  3  PQ  =  451.1  K.W.  and  335.1  K.W., 
respectively. 

9.    POLAR   CO-ORDINATES. 

In  polar  co-ordinates,  alternating  waves  are  represented, 
with  the  instantaneous  values  as  radii  vectors,  and  the  time 
as  angle,  counting  left-handed  or  counter  clockwise,  and  one 
revolution  or  360°  representing  one  complete  period. 

The  sine  wave  of  alternating  current  i  =  70  sin  <£  is 
represented  by  a  circle  (Fig.  10)  with  the  vertical  axis  as 
diameter,  equal  in  length  OIQ  to  the  maximum  value  70,  and 
shown  as  heavy  drawn  circle. 


42  ELECTRICAL   ENGINEERING. 

The  E.M.F.  consumed  by  self-induction,  r2  =  ;r/0  cos  <£, 
is  represented  by  a  circle  with  diameter  OE2  in  horizontal 
direction  to  the  right,  and  equal  in  length  to  the  maximum 
value,  xlw 

Analogously,  the  counter  E.M.F.  of  self-induction  E!  is 
represented  by  a  circle  O EJ,  in  Fig.  10,  the  E.M.F.  con- 
sumed by  resistance  r  by  circle  OE^  of  a  diameter  =  El 
=  rf0,  the  counter  E.M.F.  of  resistance  E{  by  circle  O E^. 


Fig.    10. 


The  counter  E.M.F.  of  impedance  is  represented  by 
circle  OE'  of  a  diameter  equal  in  length  to  Er,  and  lagging 
180  —  <o  behind  the  diameter  of  the  current  circle.  This 
circle  pases  through  the  points  E±  and  Ez't  since  at  the 
moment  <£  =  180°,  */  =  0,  and  thus  the  counter  E.M.F.  of 
impedance  equals  the  counter  E.M.F.  of  reactance  e'  =  e2', 
and  at  <£  =  270°,  ej  =  0,  and  the  counter  E.M.F.  of  impe- 
dance equals  the  counter  E.M.F.  of  resistance  ef  —  e{. 

The  E.M.F.  consumed  by  impedance  or  impressed  E.M.F. 


POLAR    CO-ORDINATES. 


43 


is  represented  by  circle  O  E  of  a  diameter  equal  in  length 
to  E,  and  leading  the  diameter  of  the  current  circle  by 
angle  to.  This  circle  passes  through  the  points  £l  and  E2. 

An  alternating  wave  is  determined  by  the  length  and 
direction  of  the  diameter  of  its  polar  circle.  The  length  is 
the  maximum  value  or  intensity  of  the  wave,  the  direction 
the  phase  of  the  maximum  value,  generally  called  the  phase 
of  the  wave. 

Usually  alternating  waves  are  represented  in  polar  co- 
ordinates by  mere  vectors,  the  diameters  of  their  polar 
circles,  and  the  circles  omitted,  as  in  Fig.  11. 


Two  E.M.F.'s,  ^  and  e2,  acting  in  the  same  circuit,  give  a 
resultant  E.M.F.  e  equal  to  the  sum  of  their  instantaneous 
values.  In  polar  co-ordinates  el  and  ez  are  represented  in 
intensity  and  in  phase  by  two  vectors,  O  El  and  OE2,  Fig. 
12.  The  instantaneous  values  in  any  direction  O X  are  the 
projections  Oev  O ez  of  OEl  and  OE2  upon  this  direction. 

Since  the  sum  of  the  projections  of  the  sides  of  a  paral- 
lelogram is  equal  to  the  projection  of  the  diagonal,  the  sum 
of  the  projections  Oel  and  Oe2  equals  the  projection  Oe  of 


44 


ELECTRICAL   ENGINEERING. 


OE,  the  diagonal  of  the  parallelogram  with   OEl  and 
as  sides,  and  O  E  is  thus  the  diameter  of  the  circle  of  re- 
sultant E.M.F. 

That  is, 

In  polar  co-ordinates  alternating  sine  waves  of  E.M.F., 
current,  etc.,  are  combined  and  resolved  by  the  parallelo- 
gram or  polygon  of  sine  waves. 

Since  the  effective  values  are  proportional  to  the  maxi- 
mum values,  the  former  are  generally  used  as  the  length  of 
vector  of  the  alternating  wave.  In  this  case  the  instantane- 


Fig.    12. 

ous  values  are  given  by  a  circle  with  V2  times  the  vector  as 
diameter. 

As  phase  of  the  first  quantity  considered,  as  in  the  above 
instance  the  current,  any  direction  can  be  chosen.  The 
further  quantities  are  determined  thereby  in  direction  or 
phase. 

In  polar  co-ordinates,  as  phase  of  the  current,  etc.,  is 
here  and  in  the  following  understood  the  time  or  the  angle 
of  its  vector,  that  is,  the  time  of  its  maximum  value,  and  a 
current  of  phase  zero  would  thus  be  denoted  analytically  by 
i  —  70  cos  <£. 


POLAR   CO-ORDINATES.  45 


The  zero  vector  O  A  is  generally  chosen  for  the  most 
frequently  used  quantity  or  reference  quantity,  as  for  the 
current,  if  a  number  of  E.M.F.'s  are  considered  in  a  circuit 
of  the  same  current,  or  for  the  E.M.F.,  if  a  number  of  cur- 
rents are  produced  by  the  same  E.M.F.,  or  for  the  induced 
E.M.F.  in  induction  apparatus,  as  transformers  and  induc- 
tion motors,  or  for  the  counter  E.M.F.  in  synchronous 
apparatus,  etc. 

With  the  current  as  zero  vector,  all  horizontal  compo- 
nents of  E.M.F.  are  energy  E.M.F.'s,  all  vertical  components 
are  wattless  E.M.F.'s. 

With  the  E.M.F.  as  zero  vector,  all  horizontal  components 
of  current  are  energy  currents,  all  vertical  components  of 
currents  are  wattless  currents. 

By  measurement  from  the  polar  diagram  numerical  values 
can  hardly  ever  be  derived  with  sufficient  accuracy,  since 
the  magnitude  of  the  different  quantities  entering  the  same 
diagram  is  usually  by  far  too  different,  and  the  polar  diagram 
is  therefore  useful  only  as  basis  for  trigonometrical  or  other 
calculation,  and  to  give  an  insight  into  the  mutual  relation 
of  the  different  quantities,  and  even  then  great  care  has  to 
be  taken  to  distinguish  between  the  two  equal  but  opposite 
vectors:  counter  E.M.F.  and  E.M.F.  consumed  by  the 
counter  E.M.F.,  as  explained  before. 

EXAMPLES. 

In  a  three-phase  long  distance  transmission,  the  voltage 
between  lines  at  the  receiving  end  shall  be  5000  at  no  load, 
5500  at  full  load  of  44  amperes  energy  current,  and  propor- 
tional at  intermediary  loads  ;  that  is,  5250  at  |  load,  etc. 
At  f  load  the  current  shall  be  in  phase  with  the  E.M.F.  at 
the  receiving  end.  The  generator  excitation  and  thus  the 
(nominal)  induced  E.M.F.  of  the  generator  shall  be  main- 
tained constant  at  all  loads.  The  line  has  the  resistance 
rt  =  7.6  ohms  and  the  reactance  x^  =  4.35  ohms  per  wire, 


46 


ELECTRICAL   ENGINEERING. 


the  generator  in  Y  connection  per  circuit  the  resistance 
rz  =  .71,  and  the  (synchronous)  reactance  x^  =  25  ohms. 
What  must  be  the  current  and  its  phase  relation  at  no  load, 
i  load,  £  load,  f  load,  and  full  load,  and  what  will  be  the 
terminal  voltage  of  the  generator  under  these  conditions  ? 

The  total  resistance  of  the  circuit  is,  r=  i\  +  i\  —  8.31 
ohms.     The  total  reactance,  x  =  x^  +  .rz  =  29.35  ohms. 


Fig.    13. 


Let,  in  the  polar  diagram,  Fig.  13  or  14,  O  E  =  E  repre- 
sent the  voltage  at  the  receiving  end  of  the  line,  OI^  =  1^ 
the  energy  current  corresponding  to  the  load,  in  phase  with 


Fig.    14. 


OE,  and  <9/2  =  72  the  wattless  current  in  quadrature  with 
OE,  shown  leading  in  Fig.  13,  lagging  in  Fig.  14. 

We  then  have,  total  current  1=  Of. 

Thus  E.M.F.  consumed  by  resistance,  OEl  =  rim  phase 
with  /.  E.M.F.  consumed  by  reactance,  OE^  —  xI,  90° 


POLAR    CO-ORDINATES.  47 

ahead  of  /,  and  their  resultant  the  E.M.F.  consumed  by  im- 
pedance O  Ez. 

O  Ez  combined  with  OE,  the  receiving  voltage,  gives  the 
generator  voltage  ~O~EV 

Resolving  all  E.M.F.'s  and  currents  in  components  in 
phase  and  quadrature  with  the  received  voltage  E,  we  have, 

PHASE  QUADRATURE 

COMPONENT.          COMPONENT. 

Current  1^  Iz 

E.M.F.  at  receiving  end  of  line   E  =  E  0 

E.M.F.  consumed  by  resistance  E1  =  rl^  rf2 

E.M.F.  consumed  by  reactance    £2=  xl^  —  xl± 
Thus  total  E.M.F.  or  generator  voltage 


Herein  the  wattless  lagging  current  is  assumed  as  posi- 
tive, the  leading  as  negative. 

The  generator  E.M.F.  thus  consists  of  two  components, 
which  give  the  resultant  value, 

^0  =  V(.£  H-  rl^  +  */2)2  +  (rlz  -  xfj* ; 
substituting  numerical  values,  this  becomes, 


£0  =  ^(E  +  8.31  7j  +  29.35  72)2  +  (8.31  72  -  29.35  /,)* ; 
at  |  load  it  is, 

E  =  — -?  =  3090  volts  per  circuit 
V3 

/!  =  33,         72  =  0,  thus, 

EQ  =  V(3090  +  8.31  x  33)2  +  (29.35  x  33)2  =  3520  volts 
per  line  or  3520  x  V3  =  6100  volts  between  lines, 
as  (nominal)  induced  E.M.F.  of  generator. 

Substituting  these  values,  we  have, 

3520  =  \I(E  +  8.31  7j  4-  29.35  72)2  +  (8.31  72  -  29.35  71)2. 
The  voltage  between  the  lines  at  receiving  end  shall  be : 


48  ELECTRICAL   ENGINEERING. 

AT                    No     \  \      |    FULL 

LOAD.  LOAD.  LOAD.  LOAD.  LOAD. 

Voltage  between  lines,                       5000  5125  5250  5375  5500 

Thus,  voltage  per  line,  -*-•&,£  =   2880  2950  3020  3090  3160 
The  energy  currents  per  line 

are,                                          7X  =         0       11  22       33       44 

herefrom  we  get  by  substituting  in  above  equation 

No      J      £  |        FULL 

LOAD.   LOAD.   LOAD.  LOAD.     LOAD. 

Wattless  current,  72  =          21.6     16.2        9.2  0        -    9.7 
hence,  the  total  current, 

=         21.6     19.6     23.9  33.0         45.05 
and  the  power  factor, 

4  =  cos  o>  =           0        51.0      92.0  100.0          98.0 


the  lag  of  the  current, 
<o= 

the  generator  terminal  voltage  per  line  is, 


<o  =         90°      59°       23°  0°    -  11.5° 


+  7.6  /!  +  4.35  72)2  +  (7.6  72  -  4.357J)2 
thus: 

No      £      £       2     FULL 
LOAD.   LOAD.   LOAD.   LOAD.   LOAD. 

Per  line,  E'  =  2980     3106     3228     3344     3463 

Between  lines,  Er  V3  =  5200     5400     5600     5800     6000 

that   is,  at  constant  excitation  the  generator  voltage  rises 
with  the  load,  and  proportional  thereto. 


10.    HYSTERESIS    AND    EFFECTIVE   RESISTANCE. 

If  an  alternating  current  Of  =  /,  in  Fig.  15,  passes 
through  a  circuit  of  reactance  x  =  ^^NL  and  of  negligible 
resistance,  the  magnetic  flux  produced  by  the  current,  O® 
=  <$,  is  in  phase  with  the  current,  and  the  E.M.F.  induced 


HYSTERESIS  AND  EFFECTIVE   RESISTANCE. 


49 


by  this  flux,  or  counter  E.M.F.  of  self-induction,  O  E'"  =  E'" 
=  xl,  lags  90°  behind  the  current.  The  E.M.F.  consumed 
by  self-induction  or  impressed  E.M.F.  O  E"  —  E"  =  xl  is 
thus  90°  ahead  of  the  current. 


Inversely,  if  the.  E.M.F.  O  E"  —  E"  is  impressed  upon  a 
circuit  of  reactance  x  =  2  irNL  and  of  negligible  resistance, 


<?/=/=  —  lags   90C 


behind  the  impressed 


the  current 

E.M.F. 

This  current  is  called  the  ex- 
citing or  magnetizing  current  of 
the  magnetic  circuit,  and  is  watt- 
less. 

If  the  magnetic  circuit  contains 
iron  or  other  magnetic  material, 
energy  is  consumed  in  the  mag- 
netic circuit  by  a  frictional  resist- 
ance of  the  material  against  a 
change  of  magnetism,  which  is 
called  molecular  magnetic  friction. 

If  the  alternating- current  is  the 
only  available  source  of  energy  in 
the  magnetic  circuit,  the  expendi- 
ture of  energy  by  molecular  mag- 
netic friction  appears  as  a  lag  of 
the  magnetism  behind  the  M.M.F. 
of  the  current,  that  is,  as  magnetic 
hysteresis,  and  can  be  measured 
thereby. 

Magnetic    hysteresis    is,    how- 
ever, a  distinctly  different  phenomenon  from  molecular  mag- 
netic friction,  and  can  be  more  or  less  eliminated,  as  for  in- 
stance by  mechanical  vibration,  or  can  be  increased,  without 
changing  the  molecular  magnetic  friction. 

In  consequence  of  magnetic  hysteresis,  if  an  alternating 
E.M.F.  OE"  —  E"  is  impressed  upon  a  circuit  of  negligible 


E" 


Fig.  15. 


50  ELECTRICAL   ENGINEERING. 

resistance,  the  exciting  current,  or  current  producing  the 
magnetism,  in  this  circuit  is  not  a  wattless  current,  or  current 
of  90°  lag,  as  in  Fig.  15,  but  lags  less  than  90°,  by  an  angle 
90  —  a,  as  shown  by  OI  =  /  in  Fig.  16. 

Since  the  magnetism  O&  =  $> 
is  in  quadrature  with  the  E.M.F. 
E"  due  to  it,  angle  a  is  the  phase 
difference  between  the  magnetism 
and  the  M.M.F.,  or  the  lead  of  the 
M.M.F.,  that  is,  the  exciting  cur- 
rent, before  the  magnetism.      It  is  called  the  angle  of 
hysteretic  lead. 

In  this  case  the  exciting  current  OI  =  I  can  be 
resolved  in  two  components,  the  magnetizing  current 
s//      Of2=I2  in  phase  with  the  magnetism  O$  =  3>,  that 
is,  in  quadrature  with   the  E.M.F.    OE"  =  E" ,   and 
thus  wattless,  and  the  magnetic  energy  current  or  hysteresis 
current  ~OIl  =  7lf  in  phase  with  the  E.M.F.  O E"  =  E",  or 
in  Quadrature  with  the  magnetism  O<b  —  <$. 

Magnetizing  current  and  magnetic  energy  current  are 
the  two  components  of  the  exciting  current. 

If  the  alternating  circuit  contains  besides  the  reactance 
x=  ZvNL,  a  resistance  r,  the  E.M.F.  O  E"  =  E"  in  the 
preceding  Fig.  15  and  Fig.  16  is  not  the  impressed  E.M.F., 
but  the  E.M.F.  consumed  by  self-induction  or  reactance, 
and  has  to  be  combined,  Figs.  17  and  18,  with  the  E.M.F. 
consumed  by  the  resistance,  O  Er  =  £'  =  fr,  to  get  the  im- 
pressed E.M.F.  O  E  =  E. 

Due  to  the  hysteretic  lead  a,  the  lag  of  the  current  is 
less  in  Figs.  16  and  18,  a  circuit  expending  energy  in  mole- 
cular magnetic  friction,  than  in  Figs.  15  and  17,  a  hysteresis- 
less  circuit. 

As  seen  in  Fig.  18,  in  a  circuit  whose  ohmic  resistance 
is  not  negligible,  the  magnetic  energy  current  and  the  mag- 
netizing current  are  not  in  phase  and  in  quadrature  respec- 
tively with  the  impressed  E.M.F.,  but  with  the  counter 


HYSTERESIS  AND   EFFECTIVE   RESISTANCE. 


51 


E.M.F.  of  self-induction  or  E.M.F.  consumed  by  self-induc- 
tion. 

Thus  the  magnetizing  current  is  not  quite  wattless,  as 


obvious,  since  energy  is  consumed  by  this  current  in  the 
ohmic  resistance  of  the  circuit. 

Resolving,  in  Fig.  19,  the  impressed  E.M.F.  O  E  =  E  in 
two    components,    O£l  =  E^    in    phase,   and    O  Ez  = 


n 


><*» 


quadrature  with  the  current  Of=f,  the  energy  E.M.F. 
OEl  =  El  is  greater  than  E1  —  Ir,  and  the  wattless  E.M.F. 
07?,  =  £2  less  than  E"  =  Ix. 


52  ELECTRICAL   ENGINEERING 


E,      energy  E.M.F. 
The  value  r  =  ~  =  -  -  is  called  the  effective 


I         total  current 


,      wattless  E.M.F.  . 


,  ...  . 

resistance,  and  the  value  x  =  -y  =  -     —.  —  -  is  called 

/          total  current 

the  apparent  or  effective  reactance  of  the  circuit. 

Due  to  the  loss  of  energy  by  hysteresis  (eddy  currents, 
etc.),  the  effective  resistance  differs  from,  and  is  greater 
than,  the  ohmic  resistance,  and  the  apparent  reactance  is  less 
than  the  true  or  self-inductive  reactance. 

The  loss  of  energy  by  molecular  magnetic  friction  per 
cm3  and  cycle  of  magnetism  is  approximately, 


where  (B  =  the  magnetic  induction,  in  lines  of  magnetic 
force  per  cm8, 

W=  energy,  in  absolute  units  or  ergs  per  cycle  (=  10  ~7 
watt  seconds  or  joules),  and  77  is  called  the  coefficient  of 
hysteresis. 

In  soft  annealed  sheet  iron  or  sheet  steel,  y  varies  from 
1.25  X  10~3  to  4  X  10~3,  and  can  in  average,  for  good  ma- 
terial, be  assumed  as  2.5  X  10  ~3. 

The  loss  of  power  in  volume  V,  at  magnetic  density  (B 
and  frequency  A7",  is  thus, 


X  10~7  watts 

and,  if  /=  exciting  current,  the  hysteretic  effective  resist- 

ance is, 

P  (R1-6 

>"  =  £  =  ra^io-7—- 

If  the  magnetic  induction  (B  is  proportional  to  the  cur- 
rent /,  it  is, 


that  is,  the  effective  hysteretic  resistance  is  inversely  pro- 
portional to  the  .4  power  of  the  current,  and  directly  pro- 
portional to  the  frequency. 


HYSTERESIS  AND   EFFECTIVE   RESISTANCE.  53 

Besides  hysteresis,  eddy-  or  Foucault  currents  contrib- 
ute to  the  effective  resistance. 

Since  at  constant  frequency  the  Foucault  currents  are 
proportional  to  the  magnetism  inducing  them,  and  thus 
approximately  proportional  to  the  current,  the  loss  of  power 
by  Foucault  currents  is  proportional  to  the  square  of  the 
current,  the  same  as  the  ohmic  loss,  that  is,  the  effective 
resistance  due  to  Foucault  currents  is  approximately  con- 
stant at  constant  frequency,  while  that  of  hysteresis  de- 
creases slowly  with  the  current. 

Since  the  Foucault  currents  are  proportional  to  the  fre- 
quency, their  effective  resistance  varies  with  the  square  of 
the  frequency,  while  that  of  hysteresis  varies  only  propor- 
tionally to  the  frequency. 

The  total  effective  resistance  of  an  alternating  current 
circuit  increases  with  the  frequency,  but  is  approximately 
constant,  within  a  limited  range,  at  constant  frequency,  de- 
creasing somewhat  with  the  increase  of  magnetism. 

EXAMPLES. 

A  reactive  coil  shall  give  100  volts  E.M.F.  of  self- 
induction  at  10  amperes  and  60  cycles.  The  electric  circuit 
consists  of  200  turns  (No.  8  B.  &  S.  G.)  (s  =  .013  sq.  in.)  of 
16"  mean  length  of  turn.  The  magnetic  circuit  has  a  sec- 
tion of  6  sq.  in.  and  a  mean  length  of  18",  of  iron  of  hyster- 
esis coefficient  rj  =  2.5  X  10  ~3.  An  air  gap  is  interposed 
in  the  magnetic  circuit,  of  a  section  of  10  sq.  in.  (allowing 
for  spread),  to  get  the  desired  reactance. 

How  long  must  the  air  gap  be,  and  what  is  the  resist- 
ance, the  reactance,  the  effective  resistance,  the  effective 
impedance,  and  the  power  factor  of  the  reactive  coil  ? 

200  turns  of  16"  length  and  .013  sq.  in.  section  at 
resistivity  of  copper  of  1.8  X  10~6  have  the  resistance, 

1.8  x  10-6  x  200  x  16 


r,  = 


=  .175  ohms 


.013  x  2.54 
where  the  factor  2.54  reduces  from  inches  to  cm. 


54  ELECTRICAL   ENGINEERING. 

E  =  100  volts  induced, 
N=  60  cycles,  and 
n  —  200  turns, 

the  maximum  magnetic  flux  is  given  by  E  =  4. 

or,  100  =  4.44  x  .0  x  200  $, 

as,  $=.188  ml. 

This  gives  in  an  air  gap  of  10  sq.  in.  a  maximum  density 
(B=  18,800  lines  per  sq.  in.  or  2920  lines  per  cm2. 

Ten  amperes  in  200  turns  give  2000  ampere  turns  ef- 
fective or  $  =  2830  ampere  turns  maximum. 

Neglecting  the  ampere  turns  required  by  the  iron  part 
of  the  magnetic  circuit,  2830  ampere  turns  have  to  be  con- 
sumed by  the  air  gap  of  density,  (B  =  2920. 
Since, 


the  length  of  the  air  gap  has  to  be 
4  JF        4  x  2830 


1      IOCS" 


=  1.22  cm,  or  .48". 


At  6  sq.  in.  section  and  18"  mean  length  the  volume  of 
iron  is  180  cu.  in.  or  1770  cm3. 

The  density  in  the  iron  0^=  188>QQQ  =  31330,  lines  per 

6 
sq.  in.,  or  4850  lines  per  cm2. 

At  hysteresis  coefficient  i7  =  2.5xlO~3,  and  density  (Bj  = 
4850,  the  loss  of  energy  per  cycle  and  cm3  is 

W  =  ^A1 6 

=  2.5  x  10 ~3  x  485016 
=  2220  ergs, 

and  the  hysteresis  loss  at  N  =  60  cycles  and  the  volume  V 
=  1770  is  thus, 

P=  60  X  1770  x  2220  ergs  per  sec. 
=  23.5  watts, 


CAPACITY  AND    CONDENSERS.  55 

which   at    10    amperes    represent    an    effective   hysteretic 
resistance, 

/-„  =  ~^—  =  .235  ohms. 
2       102 

Hence  the  total  effective  resistance  of  the  reactive  coil  is, 
r  =rl  +  r2  =  .175  +  .235  =  .41  ohms. 

The  reactance  is, 

p 
x  =  -'  =  10  ohms. 

Thus  the  impedance, 

s  =  10.01  ohms, 
and  the  power  factor, 

/==£==  4.1%. 

Z 

The  total  volt-amperes  of  the  reactive  coil  are, 

S2z  =  1001. 
The  loss  of  power  : 

TV  =  41. 

11.    CAPACITY   AND    CONDENSERS. 

The  charge  of  an  electric  condenser  is  proportional  to 
the  impressed  voltage  and  to  its  capacity. 

A  condenser  is  called  of  unit  capacity  if  unit  current 
flowing  into  it  during  one  second  produces  unit  difference 
of  potential  at  its  terminals. 

The  practical  unit  of  capacity  is  that  of  a  condenser  in 
which  one  ampere  flowing  during  one  second  produces  one 
volt  difference  of  potential. 

The  practical  unit  of  capacity  equals  10~9  absolute  units. 
It  is  called  a  farad. 

One  farad  is  an  extremely  large  capacity,  and  one  mil- 
lionth of  one  farad  is  commonly  used,  and  called  one  micro- 
farad, mf. 


56  ELECTRICAL   ENGINEERING. 

If  an  alternating  E.M.F.  is  impressed  upon  a  conden- 
ser, the  charge  of  the  condenser  varies  proportionally  to  the 
E.M.F.,  and  thus  current  flows  into  the  condenser  during 
rising,  out  of  the  condenser  during  decreasing  E.M.F.,  as 
shown  in  Fig.  20. 

That  is,  the  current  consumed  by  the  condenser  leads 
the  impressed  E.M.F.  by  90°,  or  a  quarter  of  a  period. 


Fig.  20. 

If  JV  =  frequency,  E  =  effective  alternating  E.M.F.  im- 
pressed upon  a  condenser  of  C.  mf  capacity,  the  condenser 
is  charged  and  discharged  twice  during  each  cycle,  and  the 

time  of  one  complete  charge  or  discharge  is  thus,  - — . 

Since  E  V2  is  the  maximum  voltage  impressed  upon 
the  condenser,  an  average  of  CE  V2  10~6  amperes  would 
have  to  flow  during  one  second  to  charge  the  condenser  to 

this  voltage,  and  to  charge4  it  during  — —  seconds,   thus  an 

average  current  of  4tNCE*J%  10~6  amperes  is  required. 

effective  current          TT 

Since  -  = -=. 

average  current       2  V2 

the  effective  current  I^-nNCE  10~6,  that  is,  at  an  impressed 
E.M.F  of  E  effective  volts  and  frequency  N,  a  condenser  of 
C  mf  capacity  consumes  a  current  of, 

/=  ZTtNCElQ-*  amperes  effective, 

which  current  leads  the  terminal  voltage  by  90°  or  a  quarter 
period,  or  inversely, 


CAPACITY  AND    CONDENSERS.  57 

106/ 

"  = 


106 
The  value  ;r0  =  —  TT^  *s  called  the  capacity  reactance  of 


the  condenser. 

Due  to  the  energy  loss  in  the  condenser  by  dielectric 
hysteresis,  the  current  leads  the  E.M.F.  by  somewhat  less 
than  90°,  and  can  be  resolved  into  a  wattless  charging  cur- 
rent and  a  dielectric  hysteresis  current,  which  latter,  how- 
ever, is  so  small  as  to  be  generally  negligible. 

The  capacity  of  one  wire  of  a  transmission  line  is, 

l.llxlO-6x/ 

~22) —      ' 

where 

d  =  diameter  of  wire,  cm, 
D  =  distance  of  wire  from  return  wire,  cm, 

/  =  length  of  wire,  cm, 
1.11  x  10~6=  reduction  coefficient  from  electrostatic  units  to  mf. 

The  logarithm  is  the  natural  logarithm,  thus  in  common 
logarithm,  since'loge  a  =  2.303  Iog10tf,  the  capacity  is, 

.25  x  10-6x/ 
C  — ^-=r —  mf . 


log 


10 


The  derivation  of  this  equation  must  be  omitted  here. 
The  charging  current  of  a  line  wire  is  thus, 


where 


JV  =  the  frequency, 

E  —  the  difference  of   potential,  effective,  between  the  line 
and  the  neutral  (E  =  ^  line  voltage  in  a  single-phase, 

or  four-wire  quarterphase  system,  —  —  line  voltage,  or 

v3 
Y  voltage,  in  a  three-phase  system). 


58  ELECTRICAL   ENGINEERING. 

EXAMPLES. 

In  the  transmission  line  discussed  in  the  examples  in 
§§  7,  8,  and  9,  what  is  the  charging  current  of  the  line  at 
6000  volts  between  lines  and  33.3  cycles?  How  many 
volt  amperes  does  it  represent,  and  what  percentage  of  the  full 
load  current  of  44  amperes  is  it  ? 

The  length  of  the  line  is,  per  wire,    /=  2.2,0  x  1C6  cm. 
The  distance  between  wires,  D  =  45  cm. 

The  diameter  of  transmission  wire,  d  =  .82  cm. 
Thus,  the  capacity,  per  wire, 


H,— 

The  frequency  is,  7^=33.3, 

The  voltage  between  lines,  6000. 

Thus  per  line,  or  between  line  and  neutral  point, 

^5™=  3460. 

V3 
Hence  the  charging  current  per  line, 


=  .195  amperes, 
or  .443%  of  full-load  current, 

that    is,    negligible    in   its   influence    on    the   transmission 
voltage. 

The  volt-ampere  input  of  the  transmission  is, 

3/o^  =  2000 

=  2.0  ATA. 

1&    IMPEDANCE   OF    TRANSMISSION    LINES. 

Let  r  =  resistance, 

x  =  2  TtNL  =  reactance  of  a  transmission  line, 
-E0=  alternating  E.M.F.  impressed  upon  the  line, 

/  =  current  flowing  over  the  line, 
E  =  E.M.F.  at  receiving  end  of  the  line,  and 

cu  =  angle  of  lag  of  current  /  behind  E.M.F.  /j. 


IMPEDANCE    OF   TRANSMISSION  LINES, 


59 


o,  <  0  thus  denotes   leading,  w  >  0  lagging  current,  and 
o,  =  0  a  noninductive  receiver  circuit. 

The  capacity  of  the  transmission  line  shall  be  considered 
as  negligible. 

Assuming  the  current  OI  =  /  as  zero  in  the  polar  dia- 
gram Fig.  21,  the  E.M.F.  E  is 
represented  by  vector  OE, 
ahead  of  OI  by  angle  w.  The 
E.M.F.  consumed  by  resistance 
r  is  OEV  =  E^  =  Ir'm  phase  with 
the  current,  and  the  E.M.F.  con- 
sumed by  reactance  x  is  OEZ 
=  E.2=fx,  90°  ahead  of  the  cur- 
rent, thus  the  total  E.M.F.  con- 
sumed by  the  line,  or  E.M.F.  consumed  by  impedance,  is 
the  resultant  O^o_f  OE^  and_a£r2,.and_is_£>8=/^. 

Combining    OE8   and    OE  gives    OEQ,  the  E.M.F.  im- 
pressed upon  the  line. 

y 

Denoting  tana  =  —  the  lag  angle  of  the  line  impedance, 
it  is,  trigonometrically, 


Fig.  21. 


OE*  = 
Since 


0  cos  OEEQ. 


we  have, 


OEEQ  =  180  -  a 


(a  -  a>), 


=  (E  +  Izf  -  ±EIz  sin2^- 


and 

£o  =  W  (E  +  Izf  —  ±EIz  sin2 

and  the  drop  of  voltage  in  the  line, 


.     a—  o> 

in2—— — 


-  E. 


60 


ELECTRICAL   ENGINEERING. 


That  is,  the  voltage  E0  required  at  the  sending  end  of  a 
line  of  resistance  r  and  reactance  x,  delivering  current  /  at 
voltage  £,  and  the  voltage  drop  in  the  line,  do  not  depend 
upon  current  and  line  constants  only,  but  depend  also  upon 
the  angle  of  phase  displacement  of  the  current  delivered 
over  the  line. 

If  w  =  0,  that  is  noninductive  circuit, 


/  jr^  4    /  /   E1       I         T  \2 

that  is  less  than  E+Iz,  and  thus  the  line  drop  less  than  Iz. 
If  w=a,  EQ  is  a  maximum,  =  E+Iz  and  the  line  drop  is 
the  impedance  voltage. 


Fig.  22. 

With  decreasing  o>,  E0  decreases,  and  becomes  =  E ; 
that  is,  no  drop  of  voltage  takes  place  in  the  line  at  a  certain 
negative  value  of  <o,  which  depends  not  only  on  s  and  a,  but 
on  E  and  /.  Beyond  this  value  of  u>,  E  becomes  smaller 


IMPEDANCE    OF   TRANSMISSION  LINES. 


61 


than  E ;  that  is,  a  rise  of  voltage  takes  place  in  the  line,  due 
to  its  reactance.     This  can  be  seen  best  graphically. 

For  the  same  E.M.F.  E  received,  but  different  phase 
angles  w,  all  vectors  OE  lie  on  a  circle  e  with  O  as  center. 
Fig.  22.  Vector  OEZ  is  constant  for  a  given  line  and  given 
current  /. 


Fig.  23. 

Since  E9E0  =  OE  =  constant,  E0  lies  on  a  circle  e0  with 
Ez  as  center  and  6>7T  =  E  as  radius. 

To  construct  the  diagram  for  angle  w,  OE  is  drawn 
under  angle  <o  with  (97,  and  EEQ  parallel  to  OEy 

The  distance  EtE0  between  the  two  circles  on  vector 
OEQ  is  the  drop  of  voltage  (or  rise  of  voltage)  in  the  line. 

As  seen  in  Fig.  23,  E0  is  maximum  in  the  direction 


62  ELECTRICAL   ENGINEERING. 

as  OEJ,  that  is  for  <o=a,  and  is  less  for  greater  as  well, 
OE^  as  smaller  angles  w.  It  is  =E  in  the  direction  O£Q' 
in  which  case  w<(9,  and  minimum  in  the  direction  OE 


Q 


The  values  of  E  corresponding  to  the  generator  voltages 
£;',  E0",  £0'",  Ei'"  are  shown  by  the  points  E'  E"  E"f  E"" 
respectively.  The  voltages  E^'  and  E^"  correspond  to  a 
wattless  receiver  circuit  E"  and  E""  .  For  noninductive 

s^pjrrrr  .        -  uiti 

receiver  circuit  OE       the  generator  voltage  is  OEQ 

That  is,  in  an  inductive  transmission  line  the  drop  of 
voltage  is  maximum  and  equal  to  Iz,  if  the  phr.se  r.rgle  ^  of 
the  receiving  circuit  equals  the  phase  angle  a  of  the  line. 
The  drop  of  voltage  in  the  line  decreases  with  increasing 
difference  between  the  phase  angles  of  line  and  receiving 
circuit.  It  becomes  zero  if  the  phase  angle  of  the  receiving 
circuit  reaches  a  certain  negative  value  (leading  current). 
In  this  case  no  drop  of  voltage  takes  place  in  the  line.  If 
the  current  in  the  receiving  circuit  leads  more  than  this 
value  a  rise  of  voltage  takes  place  in  the  line.  Thus  by 
varying  phase  angle  w  of  the  receiving  circuit,  the  drop  of 
voltage  in  a  transmission  line  with  current  /  can  be  made 
anything  between  Iz  and  a  certain  negative  value.  Or  in- 
versely the  same  drop  of  voltage  can  be  produced  for  differ- 
ent values  of  the  current  /  by  varying  the  phase  angle. 

Thus,  if  means  are  provided  to  vary  the  phase  angle  of 
the  receiving  circuit,  by  producing  lagging  and  leading 
currents  at  will  (as  can  be  done  by  synchronous  motors  cr 
converters)  the  voltage  at  the  receiving  circuit  can  te  main- 
tained constant  within  a  certain  range  irrespective  of  the 
load  and  generator  voltage. 

In  Fig.  24  let  OE=E  the  receiving  voltage,  /  the  energy 
current  flowing  over  the  line,  thus  OEZ=EZ=I^  the  E.M.F. 
consumed  by  the  impedance  of  the  energy  current  consist- 
ing of  the  E.M.F.  consumed  by  resistance  OE^  and  the 
E.M.F.  consumed  by  reactance  OEy 

Wattless  currents  are  represented  in  the  diagram  in  the 
direction  OA  when  lagging  and  OB  when  leading.  The 


IMPEDANCE    OF   TRANSMISSION  LINES. 


63 


E.M.F.  consumed  by  impedance  of  these  wattless  currents  is 
thus  in  the  direction  e^  at  right  angles  to  OEZ.  Combining 
OE3  and  OE  gives  the  E.M.F.  OEV  which  would  be  required 
for  noninductive  load.  If  EQ  is  the  generator  voltage,  ^0-lies 
on  a  circle  r  with  OEQ  as  radius.  Thus  drawing  E^EQ  par- 


' .  B 

Fig.  24. 

allel  to  el  gives  OE9  the  generator  voltage, 
th-2  E.M.F.  consumed  by  impedance  of  the  wattless  current, 
and  proportional  thereto  OI'=Ir  the  wattless  current  re- 
quired to  give  at  generator  voltage  EQ  and  energy  current  /, 
the  receiver  voltage  E.  This  wattless  current  I'  lags  be- 
hind El  by  less  than  90  and  more  than  zero  degrees. 

In  calculating  numerical  values,  ws  can  either  proceed 
trigonometrically  as  in  the  preceding,  or  algebraically  by 


64 


ELECTRICAL   ENGINEERING. 


resolving  all  sine  waves  in  two  rectangular  components,  for 
instance,  a  horizontal  and  vertical  component,  in  the  same 
way  as  in  mechanics  when  combining  forces. 

Let  the  horizontal  components  be  counted  positive 
towards  the  right,  negative  towards  the  left,  and  the  verti- 
cal component  positive  upwards,  negative  downwards. 

Assuming  the  receiving  voltage  as  zero  line  or  positive 
horizontal  line,  the  energy  current  /  is  the  horizontal,  the 
wattless  current  /'  the  vertical  component  of  current.  The 
E.M.F.  consumed  by  resistance  is  horizontal  component  with 
the  energy  current  /  and  vertical  component  with  the  watt- 
less current  /',  and  inversely  the  E.M.F.  consumed  by 
reactance. 

We  have  thus,  as  seen  from  Fig.  24. 


HORIZONTAL 
COMPONENT. 


Receiver  voltage  7s, 

Energy  current,  7, 

Wattless  current,  7', 

E.M.F.  consumed  by  resistance  r 
of  energy  current,  Ir, 

E.M.F.  consumed  by  resistance  r 
of  wattless  current  I'r, 

E.M.F.  consumed  by  reactance  x 
of  energy  current,  7r, 

E.M.F.  consumed  by  reactance  x 
of  wattless  current,  I'x, 

Thus,  total  E.M.F.  required  or  im- 
pressed E.M.F.,  T£O, 

hence,  conbined, 

£0  = 
or  expanded, 


-\-E 
-f-/ 
0 


VERTICAL 
COMPONENT- 

0 
0 

±/' 


+  Ir 

.   0 

0 

±I'r 

0 

-Ix 

±I'x 

0 

fr±f'x      ±  I'r  -  Ix, 


Ir  ±  I'xf  +  (±f'r- 


2£(fr  ±  I'x)  +  (72  +  //2)  s?. 

From  this  equation  /'  can  be  calculated,  that  is,  the  watt- 
less current  found  which  is  required  to  give  EQ  and  E  at 
energy  current  7. 


IMPEDANCE    OF    TRANSMISSION  LINES.  65 

The  lag  of  the  total  current  in  the  receiver  circuit  behind 
the  receiver  voltage  is 


tan  o)  =  — 


The  lead  of  the  generator  voltage  ahead  of  the  receiver 
voltage  is 


vertical  component  of  EQ 
tan  6  =  -  —  :  --  =  —  —  7—=^ 

horizon  al  component  of  Eo 

±  Ifr  -  Ix 
~  E  +  Ir  ±  /'#' 

and  the  lag  of  the  total  current  behind  the  generator  volt- 
age is 

0)0  =    OJ   +   0. 

As  seen,  by  resolving  into  rectangular  components  the 
phase  angles  are  directly  determined  from  these  components. 

The  resistance  voltage  is  the  same  component  as  the 
current  to  which  it  refers. 

The  reactance  voltage  is  a  component  90°  ahead  of  the 
current. 

The  same  investigation  as  made  here  on  long-distance 
transmission  applies  also  to  distribution  lines,  reactive  coils, 
transformers,  or  any  other  apparatus  containing  resistance 
and  reactance  inserted  in  series  into  an  alternating  current 
circuit. 

EXAMPLES. 

1.  In  an  induction  motor,  at  2000  volts  impressed  upon 
its  terminals,  the  current  and  the  power  factor,  that  is,  the 
cosine  of  the  angle  of  lag,  as  function  of  the  output,  are  as 
given  in  Fig.  25. 

The  induction  motor  is  supplied  over  a  line  of  resistance 
^=2.0  and  reactance  ;r=4.0. 

a)  How  must  the  generator  voltage  e0  be  varied  to  main. 
tain  constant  voltage  ^=2000  at  the  motor  terminals,  and 

b)  At  constant  generator  voltage  <?0=2300,  how  will  the 
voltage  at  the  motor  terminals  vary. 


66  ELECTRICAL    EATGINEERING. 

We  have, 


•  »     /  / 

eo  =  V 

2  = 


•      o 

/J"  ~      "*  Slrr 


=  4.472 


tan  a  =  -  =  2. 
r 


e  =  2000. 


a  =  63.4°. 


cos  o>  =  power  factor, 

and  /'  taken  from  Fig.  25  and  substituted,  gives  a)  the  values 
of  e0  for  <?=2000,  recorded  on  attached  table,  and  plotted  in 
Fig.  25. 


10  20  30  40  50  CO  70  SO  90  100  110  120  130  110  IM  100  170  130  190  200 
Fig.  25. 

b)  At  generator  voltage  eQ)  the  terminal  voltage  of  motor 
is  ^=2000,  the  current  z,  the  output  P.  Thus  at  generator 
voltage  <r0'=2300,  the  terminal  voltage  of  motor  is, 


the  current, 
and  the  power, 


*0  ^0 

2300  . 


IMPEDANCE    OF   TRANSMISSION  LINES. 


67 


The  values  of  e'y  if,  P'  are  recorded  in  the  second  part 
of  attached  table  under  b)  and  plotted  on  Fig.  26. 


K.W. 
150 


130 


Volts 
2100 


no 


2000 


100 


7 


90| 


Amps. 


20        3J        10        50 


70        <">        90 
Fig.  26. 


100      110      120       130      liO 


a)   AT*  =  2000, 

THUS, 
<?0. 

V)   HENCE,  AT  e0  =  2300, 

OUTPUT, 
P  —  KW. 

CURRENT, 
(. 

LAG, 

W. 

OUTPUT, 
P. 

CURRENT, 
if. 

VOLTAGE, 
e'. 

0 

12.0 

84.3° 

2048 

0. 

13.45 

2240 

5 

12.6 

726° 

2055 

6.25 

14.05 

2234 

10 

13.5 

620° 

2060 

12.4 

'  15.00 

2230 

15 

148 

54.6° 

2065 

18.6 

16.4 

2220 

20 

1(5.3 

47.9° 

2071 

24.4 

18.0 

2216 

30 

20.0 

378° 

2084 

36.3 

22.0 

2200 

40 

25.0 

32.8° 

2093 

48.0 

275 

2198 

50 

30.0 

29.0° 

2110 

59.5 

32.7 

2180 

69 

40.0 

26  3° 

2146 

78.5 

42.8 

2160 

102 

63.0 

245° 

2216 

110.2 

62.6 

20SO 

132 

80.0 

25  8° 

2294 

131.0 

79.5 

199:) 

160 

100.0 

28.4° 

2382 

149.0 

96.4 

1928 

180 

120.0 

31.8° 

2476 

156.5 

111.5 

1800 

2CO 

150.0 

36.9° 

2618 

155. 

mo 

1760 

68 


ELECTRICAL   ENGINEERING, 


2.)  Over  a  line  of  resistance  r=2.0  and  reactance  x= 
6.0,  power  is  supplied  to  a  receiving  circuit  at  constant  volt- 
age of  *r  =  2000.  How  must  the  voltage  at  the  beginning  of 
the  line,  or  generator  voltage,  eQ  be  varied,  if  at  no  load  the 


Energy 


0     20     30 


110    120    1  0  140    150  1 


0   170  180   190  200 


Fig.  27. 

receiving  circuit  consumes  a  wattless  current  of  /2=  20  am- 
peres, this  wattless  current  decreases  with  the  increase  of 
load,  that  is,  of  energy  current  iv  becomes  t'2=Q  at  z'i  =  50 
amperes,  and  then  increases  again  at  the  same  rate  as  lead- 
ing current  ? 

The  wattless  current, 

/i  =  20  at  /j  =    0, 
i2=    0  at  /i  =  50, 

and  can  be  represented  by, 


the  general  equation  of  the  transmission  line  is, 


=  V(20'00 
p  EL     1473     43 


(2  /;  -  6  ^ 


ALTERNATING    CURRENT    TRANSFORMER.  69 

hence,  substituting  the  value  of  iv 

e0  =  V(2120  -Ai\y  +  (40  -  6. 


=  V4,496,000  +  46.4  ?12  -  2240  t\. 

Substituting  successive  numerical  values  for  iv  gives  the 
values  recorded  on  attached  table,  and  plotted  in  Fig.  27. 


*'r 

g  , 

0 

2120 

20 

2116 

40 

2112 

60 

2128 

80 

21C8 

100 

2176 

120 

2213 

140 

2256 

160 

2308 

180 

2365 

200 

2430 

13.    ALTERNATING    CURRENT    TRANSFORMER. 

The  alternating  current  transformer  consists  of  one  mag- 
netic current  interlinked  with  two  electric  circuits,  the  pri- 
mary circuit  which  receives  energy,  and  the  secondary  circuit 
which  delivers  energy. 

Let  /-^resistance,  x^=  2irNS1  =  self  -inductive  reactance 
of  secondary  circuit. 

r0— resistance,  x^=  2irNS0= self  -inductive  reactance  of  pri- 
mary circuit. 

Where  Sl  and  S0  refer  to  that  magnetic  flux  which  is 
interlinked  with  the  one,  but  not  with  the  other  circuit. 

,        .       .  secondary  .        . 

Let  #=equal  ratio  of  -  — -    turns  (ratio    of  trans- 

primary 

formation). 

An  alternating  E.M.F.  £0  impressed  upon  the  primary 
electric  current  causes  a  current  to  flow,  which  produces  a 
magnetic  flux  $  interlinked  with  primary  and  secondary  cir- 
cuit. This  flux  3>  induces  E.M.F.'s  E^  and  E{  in  secondary 


70  ELECTRICAL   ENGINEERING. 

and  in  primary  circuit,  proportional  with  each  other  by  the 
£ 

ratio  of  turns,  E{  =— l. 
a 

jLet  E= secondary  terminal  voltage,  /^secondary  cur- 
rent, w!  =  lag  of  current  1^  behind  terminal  voltage  E  (where 
Wj  <  0  denotes  leading  current). 

Denoting  then  in  Fig.  28  by  a  vector  OE=E  the  sec- 
ondary terminal  voltage,  Ol^=I^  is  the  secondary  current 
lagging  by  angle 


Fig.  28. 


The  E.M.F.  consumed  by  the  secondary  resistance  i\  is 
{=Ei=Ifi  in  phase  with  7r 

The  E.M.F.  cons-Limed  by  the  secondary  reactance  x^  is 
"=E"=IT?V  90°  ahead  of  7r     Thus  the  E.M.F.   con- 


sumed by  the  secondary  impedance  s^=  V^2-!-  A\2  is  the  re- 
sultant of  OE{  and  OEf,  or  OE  "'  =  £{"=!&._ 

OE  i"  combined  with  the  terminal  voltage  OE=E  gives 
the  secondary  induced  E.M.F.  OE^=-EV 

Proportional  thereto  by  the  ratio  of  turns  and  in  phase 

therewith  is  the  E.M.F.   induced-Jn  the  .primary  OEi=Ei 

E 

where  £".=—!. 
a 

To  induce  E.M.F.  El  and  Eiy  the  magnetic  flux  O&  =  ®  is 
required,  90°  ahead  of  OE^  and  OE^  To  produce  flux  3>  the 


ALTERNATING    CURRENT   TRANSFORMER.  71 

M.M.F.  of  CF  ampere  turns  is  required,  as  determined  from 
the  dimensions  of  the  magnetic  circuit,  and  thus  the  primary 
current  Iw  represented  by  vector  OIW  leading  O$>  by  angle 
a,  the  angle  of  hysteretic  advance  of  phase. 

Since  the  total  M.M.F.  of  the  transformer  is  given  by 
the  primary  exciting  current  Iw  corresponding  to  the  sec- 
ondary current  7X  a  component  of  primary  current  /',  which 
may  be  called  the  primary  load  current,  flows  opposite  thereto 
and  of  the  same  M.M.F.,  that  is,  of  the  intensity  I'=alv  thus 
represented  by  vector  OI'  =  r=aIl. 

OJm,  the  primary  exciting  current,  and  the  primary  load 
current  (97',  or  component  of  primary  current  corresponding 
to  the  secondary  current,  combined,  give  the  total  primary 
current  C?/0=/0. 

The  E.MiF.  consumed  by  resistance  in  the  primary  is 
OE£  =  EQ  —  /0"o  in  phase  with  70. 

The  E.M.F.  consumed  by  the  primary  reactance  is  OEQ" 
=  E?=Ijc»  90°jihead  of  OIV 

OEJ  and  OE^  combined  give  OE0'",  the  E.M.F.  con- 
sumed by  the  primary  impedance. 

Equal  and  opposite  jto  the  primary  induced  E.  M.F.  OEt 
is  the  component  of  primary  E.M.F.  consumed  thereby  OE'  . 


OE'  combined  with  OE"'  gives  OEQ  =  EQ,  the  primary 
impressed  E.M.F.,  and  angle  u>Q=EQ  CIQ)  the  phase  angle  of 
the  primary  circuit. 

Figs.  29,  30,  and  31  give  the  polar  diagrams  for  u^  —  450 
or  lagging  current,  w^zero  or  noninductive  circuit,  and  0^  = 
—  45°  or  leading  current. 

As  seen,  the  primary  impressed  E.M.F.  EQ  required  to 
produce  the  same  secondary  terminal  voltage  E  at  the  same 
current  7X  is  larger  with  lagging  cr  inductive,  smaller  with 
leading  current  than  on  a  noninductive  secondary  circuit,  or 
inversely  at  the  same  secondary  current  7X  the  secondary 
terminal  voltage  E  with  lagging  current  is  less,  and  with 
leading  current  more  than  with  noninductive  secondary 
circuit,  at  the  same  primary  impressed  E.M.F.  7T0. 


72 


ELECTRICAL   ENGINEERING. 


The  calculation  of  numerical  values  is  not  practicable  by 
measurement  from  the  diagram,  since  the  magnitudes  of 
the  different  quantities  are  too  different.  E{  :  E"  :  El :  E9 
being  frequently  in  the  proportion  1  : 10  :  100  :  2000. 


Fig.  29. 


Trigonometrically,  the  calculation  is  thus : 
In  triangle  OEEV  Fig.  28,  we  have, 
writing, 

tan  ^  =  — , 


also, 


hence, 


OE*  =  O£2  +  ££l  -  2 


cos 


V  =  180  -  ^  + 
£*=&  +  I?z?  + 


cos    <     - 


This  gives  the  secondary  induced  E.M.F.,  Ev  and  there- 
from the  primary  induced  E.M.F. 


ALTERNATING   CURRENT   TRANSFORMER.  *  73 

In  triangle  EOEV  we  have, 

sin  El  OE  -f-  sin  E^EO  =  EEl  -f-  ^O; 
thus,  writing 

We  have, 

sin  ft  -f-sin  (^  — wj)  =  I^z  -f-  £v 


Fig.  30. 


r.  37. 


therefrom  we  get 


A  and 


ft, 


the  phase  displacement  between  secondary  current  and  sec- 
ondary induced  E.M^F. 

In  triangle,  OI^  we  have, 


cos  O/ 


74  ELECTRICAL    ENGINEERING. 

since 

^Cty  =  90°, 

4.   0/00/0=90-o>  -a, 

and 

^70  =  /'=«/;. 

O/oo  =  /^  =  exciting  current, 

calculated  from    the    dimensions  of    the  magnetic    circuit, 
Thus, 

/02  =  /^  +  a2//  +  2  a/j/oo  sin  (<o  +  a), 

the  primary  current. 

In  triangle  Of0  /0,  we  have 

sin  /ooO/o  -r-  sin  O^/0  =  /^  -s-  O70, 
writing 

^  ^oo  ^^o  —  £o 

this  becomes 

sin  y80  ^-  sin  (o>  -f  a)  =  a/j  +  70  ; 

therefrom  we  get  /30,  and  thus 


In  triangle  OE'  EQ  we  have 


*  f2         '* 


OE    =  0£f+  E'E    -  2  OE'E'Es  cos  OE'  E 


writing 
we  have 


tan  <>  =  -° 


0^^0=180-^0 


E'EQ  = 
thus, 


=        +  /.V  +  cos  (*,  -  ft, 


the  impressed  E.M.F. 
In  triangle,  OE'  EQ  is 

sin  E'  OEQ  +  sin  OE'EQ  = 


ALTERNATING    CURRENT  TRANSFORMER.  75 

thus,  writing, 

ZE'OE,=  y, 
we  have 

sin  y  -f-  sin  (<£0  -  <£)  =  /02"0  H-  j5"0  ; 

herefrom  we  get  4-  y,  and 

•*  "o  =  00+  T' 

the  phase  displacement  between  primary  current  and  im- 
pressed E.M.F. 

As  seen,  the  trigonometric  method  of  transformer  calcu- 
lation is  rather  complicated. 

Somewhat  simpler  is  the  algebraic  method,  of  resolving 
into  rectangular  components. 

Considering  first  the  secondary  circuit,  of  current  1^  lag- 
ging behind  the  terminal  voltage  E  by  angle  o>r 

The  terminal  voltage  E  has  the  components  E  cos  o^  in 
phase,  E  sin  Wj  in  quadrature  with  and  ahead  of  the  current  Iv 

The  E.M.F.  consumed  by  resistance  rv  Ij\,  is  in  phase. 

The  E.M.F.  consumed  by  reactance  jrp  7rrp  is  in  quad- 
rature ahead  of  7r 

Thus  the  induced  secondary  E.M.F.  has  the  components, 

E  costy-f  Ij\  in  phase, 

E  sin  w1  +  Ilxl  in  quadrature  ahead  of  the  current  flt 

and  the  total  value  : 


EI  =  VCE  cos  *>!  -f  71r1)2  +  (E  sin  Wl  -f  /i 
and  the  phase  angle  of  the  secondary  circuit  is  : 

'E  sin  <*>!  +  Ilxl 


tan  o>  = 


E  COS 


Resolving  all  quantities  into  components  in  phase  and  in 
quadrature  with  the  induced  E.M.F.  Ev  or  in  horizontal  and 
in  vertical  components,  chosing  the  magnetism  or  mutual 
flux  as  vertical  axis,  and  denoting  the  direction  to  the  right 
and  upwards  as  positive,  to  the  left  and  downwards  as  nega- 
tive, we  have, 


76  ELECTRICAL   ENGINEERING. 

HORIZONTAL         VERTICAL 
COMPONENT.      COMPONENT. 

Secondary  current  7V  —  1^  cos  w     —  I±  sin  o> 

Seecondary  induced  E.M.F.,  £v  —  E{  0 

Primary  induced,  or  counter,  E.M.F., 

a  '  a 

Primary    E.M.F.    consumed    thereby, 


Primary  load  current,  Jf=—afl  -+ 

Magnetic  flux  $,  0  <I> 

Primary  exciting  current,  7^,  consist- 
ing of  hysteresis  current,  f^  sin  a 
Magnetizing  current,  1^  cos  a 

Hence,  total  primary  current,  /0, 

HORIZONTAL  COMPONENT.  VERTICAL  COMPONENT. 

#/!  cos  <«>!  +  /oo  sin  a  alv  sin  Wj  -f-  /oo  cos  a 

E.M.F.  consumed  by  primary  resistance  r0,  E^  —  I^  in  phase 
with  7^, 

HORIZONTAL  COMPONENT.  VERTICAL  COMPONENT. 

r0^^  cos  o>  •+-  r0^Qsin  a         rtfi-[\  sin  w  -f~  ^'0^00  cos  « 

E.M.F.  consumed   by  primary  reactance   XQ,  £0=  f^,    90° 
ahead  of  IQ, 

HORIZONTAL  COMPONENT.         VERTICAL  COMPONENT. 

\  sin  o>  -f-  #0/00  cos  a      —  ^0^/j  cos  a>  —  #0/00  sin  a 


E.M.F.  consumed  by  primary  induced  E.M.F.,  E'  =  — , 
Thus,  total  primary  impressed  E.M.F.,  JS0. 

HORIZONTAL  COMPONENT. 

F 

-  -f-  e^  (r0  cos  w  -h  ^0  sin  w)  -f-  7^  (r0  sin  a  +  ^0  cos  a). 

VERTICAL  COMPONENT. 

al±  (rQ  sin  <•>  —  XQ  cos  w)  +  /»(/o  cos  a  —  ^Osin  a), 

X 

or  writing,  tan  <£0  =  — °5 

since 


ALTERNATING    CURRENT   TRANSFORMER.  77 

substituting  this  value,  horizontal  component  of  EQJ 

E, 

h  azQ/^cos  (w  —  <£0)  +  So/oo  sm  (a  +  ^0)5 

(i 

vertical  component  of  E0, 

aZql^  sin  (o>  —  ^>0)  -|-  ^^o  cos  (a  +  <^0)) 
and,  total  primary  impressed  E.M.F., 

/ 1  A\  T^    j  T» 

E({=.\  I \-UZQ! ^ cos  («j — 00)-[-2:0/00 sin  (a-f- </>0)  I  -j-l  az0/iSin(ia — 4»0) -}~ ZQ!<M  cos  (a ~r 4*o)  I 

*    LI 


The    total    primary    current    is,  by  combining    its   two 
components, 


fQ  =      (<ar^  cos  to  -f-  7^  sin  a)2  +  (0/L  sin  o>  •+•  1^  cos  a) 


/oo2 


Since  the  tangent  of  the  phase  angle  is  the  ratio  of 
vertical  component  to  horizontal  component,  we  have, 
primary  E.M.F.  phase, 

tan     =      az^  Sin  (°*  ~  ^  +  ;2'0/00  C°S  ^a  +  ^      ; 
— L  +  az^  cOs  (w  —  <£0)  +  2-0/oo  sin  (a  —  <£0) 

primary  current  phase, 

a/,  sin  o>  4-  /x)  cos  a 

tan  ^  =  -J—  ,,00   . — , 

rt/1cosw  +  /ooSin  a 

and  lag  of  primary  current  behind  impressed  E.M.F., 

wo  =  ^  —  X- 

EXAMPLES. 

1).    In  a  20  K.W.  transformer   the  ratio  of  turns  is, 
20  -s-  1,  and  100  volts  is  produced  at  the  secondary  terminals  . 
at  full  load.     What  is  the  primary  current  at  full  load,  and 


78 


ELE  C  TRIG  A  L  ENGINEERING. 


the  regulation,  that  is,  the  rise   of  secondary  voltage  from 

full   load  to  no  load,  at  constant  primary  voltage,  and  what 

is  this  primary  voltage  ? 

a),    at  noninductive  secondary  load, 

b).    with  60°  lag  in  the  external  secondary  circuit, 

c).    with  60°  lead  in  the  external  secondary  circuit. 

The  exciting  .current   is  .5  amperes,  the  hysteresis  loss 

600    watts,  the    primary    resistance   2    ohms,    the    primary 

reactance  5  ohms,  the   secondary  resistance  .004  ohms,  the 

secondary  reactance  .01  ohm. 

600  watts    at    2000   volts    give    .3    amperes    magnetic 

energy  current,  hence   V.52—  32  =  .4  amperes  magnetizing 

current. 

We  have  thus, 

a  =  .05 


2 

rt  =  .004 

TOO  COS  a  =.3 

5 

*x  =  .01 

^o  sin  a  =  .4 

/oo  =  .5 

1.    Secondary  current  as  horizontal  axis,  — 


NONINDUCTIVE, 

U>t=0. 

LAG, 
•^=£  +  60°. 

LEAD, 
u>!  =  —  60°. 

HOR. 

VERT. 

HOR. 

VERT. 

HOR. 

VERT. 

Secondary  current, 
7i 

200 

100 
.8 
0 
100.8 

0 

0 
0 
-2.0 
-2.0 

200 
50 
.8 
0 
50.8 

0 

-86.6 
0 
-2.0 
-88.6 

200 
50 

.8 
0 
50.8 

0 

+  866 
0 
-2.0 
+J84.6 

Secondary  terminal 
voltage,  E  .     .     . 
Resistance  voltage, 
Tin    .... 

Reactance  voltage, 

/!#» 

Secondary  induced 
E.M.F.,^i.     .     . 

Secondary  induced 
E.M.F.,  total  .     . 
tan  w    
i>) 

10080 
+  .0198 
+  1.1° 

102.13 
+  1.745 
+  60.2° 

98.68 
-  1.665 
-  59.0° 

RE  C  TANG  ULAR    CO-ORDINA  TES. 

2.    Magnetic  flux  as  vertical  axis,  — 


79 


NON  INDUCTIVE. 
<»!  —  0. 

LAG, 
u,1  =  +  60°. 

LEAD. 
Wl  =  —  60°. 

HOR. 

VERT. 

HOR. 

VERT. 

HOR. 

VERT. 

Secondary  induced 

E.M.F.,  £i      .     . 

-100.80 

0 

-102.13 

0 

-  98.68 

0 

Secondary  current, 

7i  

-200 

-4 

-99.4 

-172.8 

-103 

+  171.4 

Primary  load   cur- 

current,  I'—  —  a/1 

+  10 

+  .2 

+  4.97 

+  8.64 

+  5.15 

-8.57 

Primary      exciting 

current,  Too      .     . 

.3 

.4 

.3 

.4 

.3 

.4 

Total  primary  cur- 

rent, 7o  .     .     .     . 

+  10.3 

+  .6 

+  5.27 

+  9.04 

+  5.45 

-8.17 

Primary  resistance, 

voltage,  /oro    .     . 

20.6 

1.2 

10.54 

18.C8 

10.90 

-  16.34 

Primary  reactance, 

voltage,  Aro  .     . 

3.0 

-51.3 

45.20 

-26.35 

-40.85 

-  27.25 

E.M.F.    consumed 

by  primary  coun- 

ter E.M.F.,—  . 

2016 

0 

2042.6 

0 

1973.6 

0 

a 

Total  primary  im- 

pressed E.M.F.  £n 

2039.6 

-50.1 

2098.34 

-8.27 

1943.65 

-  43.59 

Hence,  — 


• 

NONINDUCTIVE, 
Wl=0. 

LAG, 
wt  =  +  60°. 

LEAD, 
u>t=—  60°. 

Resultant  .£"0       

20401 

2008  3 

19442 

Resultant  /<  

1032 

1047 

982 

Phase  of  £o        

-1.4° 

-.2° 

-1.2° 

Phase  of  7o         . 

+  33° 

+  598° 

—  50  30 

Primary  lag,  wo 

+  47° 

+  600° 

—  551° 

Regulation^^      .... 

1.02005 

1.04915 

.972 

Drop  of  voltage,  per  cent    . 

2.C05 

4.C15 

-2.79 

Change  of  phase,  wo  —  on 

4.7° 

0 

4.9° 

14.     RECTANGULAR   CO-ORDINATES. 

The  polar  diagram  of  sine  waves  gives  the  best  insight 
into  the  mutual  relations  of  alternating  currents  and 
E.M.F.'s. 

For  numerical  calculation  from  the  polar  diagram  either 


80  ELECTRICAL   ENGINEERING. 

the  trigonometric  method  or  the  method  of  rectangular 
components  is  used. 

The  method  of  rectangular  components,  as  explained  in 
the  last  paragraphs,  is  usually  simpler  and  more  convenient 
than  the  trigonometric  method. 

In  the  method  of  rectangular  components  it  is  desirable 
to  distinguish  the  two  components  from  each  other  and  from 
the  resultant  or  total  value  by  their  notation. 

To  distinguish  the  components  from  the  resultant,  small 
letters  are  used  for  the  components,  capitals  for  the  result- 
ant. Thus  in  the  transformer  diagram  of  section  13  the 
secondary  current  7X  has  the  horizontal  component  j\  =  —  7j 
cos  Wj,  and  the  vertical  component  z/  =  —  /x  sin  o>r 

To  distinguish  horizontal  and  vertical  components  from 
each  other,  either  different  types  of  letters  can  be  used,  or 
indices,  or  a  prefix  or.  coefficient. 

Different  types  of  letters  are  inconvenient,  indices  dis- 
tinguishing the  components  undesirable,  since  indices  are 
reserved  for  distinguishing  different  E.M.F.'s,  currents,  etc., 
from  each  other. 

Thus  the  most  convenient  way  is  the  addition  of  a  pre- 
fix or  coefficient  to  one  of  the  components,  and  as  such  the 
letter/  is  commonly  used  with  the  vertical  component. 

Thus  the  secondary  current  in  the  transformer  diagram 
section  13  can  be  written, 


=  I\  cos  °>i  +/A  sn  °V' 

This  method  offers  the  further  advantage,  that  the  two 
components  can  be  written  side  by  side,  with  the  plus  sign 
between  them,  since  the  addition  of  the  prefix/  distinguishes 
the  value  yV2  or//!  sinfcJj  as  vertical  component  from  the  hori- 
zontal component  ^  or  /x  cos  <or 


thus  means,  Ix  consists  of  a  horizontal  component  ^  and  a 
vertical  component  z'2,  and  the  plus  sign  signifies  that  ^  and 
?2  are  combined  by  the  parellelogram  of  sine  waves. 


RECTANGULAR    CO-ORDINATES.  81 

The  secondary  induced  E.M.F.  of  the  transformer  in 
section  13,  Fig.  28,  is  written  in  this  manner,  El—  —  ev 
that  is,  it  has  the  horizontal  component  —  ^  and  no  vertical 
component. 

The  primary  induced  E.M.F.  is, 

£,=  —  S 
a 

and  the  E.  M.  F.  consumed  thereby 


The  secondary  current  is, 

fi  =  ~h- 
Where 

t\  =  fi  COS  (I)},     4  = 

and  the  primary  load  current  corresponding  thereto  is, 


The  primary  exciting  current, 

/oo=^+y^ 

where  h  =  f^  sin  a  is  the  magnetic  energy  current,  g  = 
cos  a  the  wattless  magnetizing  current. 
Thus  the  total  primary  current  is, 

/o  =  /'  +  /oo  =  K  +  K)  +j(<*i*  +  ^). 
The  E.M.F.  consumed  by  primary  resistance  r0  is 


The  horizontal  component  of  primary  current  (at\+/i) 
gives  as  E.M.F.  consumed  by  reactance  XQ  a  negative  vertical 
component,  denoted  by  —  y>0  (rt/j  +  //).  The  vertical  com- 
ponent of  primary  current/  (aiz  +  g)  gives  as  E.M.F.  con- 
sumed by  reactance  ^TO  a  positive  hoizontal  component, 
denoted  by  *0  («/2'  +  ^). 

Thus  the  total  E.M.F.  consumed  by  primary  reactance 
;r0is, 

•*o  («4  4-  ^)  ~>o  («4  +  >*)» 


82  ELECTRICAL   ENGINEERING. 

and  the  total  E.M.F.  consumed  by  primary  impedance  is, 
[r0  (04  +  //)  +  XQ  (at's  +  £)]  +/  [r0  (>2  +  £>  -  *0  (at\  +  /?)]. 

Thus,  to  get  from  the  current  the  E.M.F.  consumed  by 
reactance  x^  to  the  horizontal  component  of  current  the 
coefficient  —j  has  to  be  added,  in  the  vertical  component  the 
coefficient  j  omitted.  Or  we  can  say, 

The  reactance  is  denoted  by  —  jxQ  for  the  horizontal, 

and  by  '-^  for  the  vertical  component  of  current. 

Or  other  words, 

If    /=  i  +jir  is  a  current,  x  the  reactance  of  its  circuit, 
the  E.M.F.  consumed  by  the  reactance  is, 

—  jxi  +  xir  =  xir  —  jxi. 

If  instead  of  emitting^'  in  deriving  the  reactance  E.M.F. 
for  the  vertical  component  of  current,  we  would  add  —  yalso, 
(as  done  when  deriving  the  reactance  E.M.F.  for  the  hori- 
zontal component  of  current)  we  get  the  reactance  E.M.F. 

—jxi—fxi', 
which  gives  the  correct  value  —jxi  +  xi',  if 

i2  —          1 

J  •*•) 

that  is,  we  can  say, 

In  deriving  the  E.M.F.  consumed  by  reactance,  x^  from 
the  current,  we  multiply  the  current  by  —  jx,  and  substitute 

j\=  - 1- 

— -jx  can  thus  be  called  the  reactance  in  the  representa- 
tion in  rectangular  co-ordinates  and 

r  —  jx  the  impedance, 
by  defining,  and  substituting, 

/>  =  -!. 

The  primary  impedance  voltage  of  the  transformer  in 
the  preceding  could  thus  be  derived  directly  by  multiplying 
the  current, 


RECTANGULAR    CO-ORDINATES. 

with  the  impedance 


83 


+  J  (<*** 


This  gives 

£Q'=  zofo  =  (fo 

=    >0  (<«1  + 

and  substituting/2  =  —  1, 

£0  =  [r0  (at\  +  /i)  +  ^0  (<*t*  + 

and  the  total  primary  impressed  E.M.F.  is  thus, 


"*  f  *   ™  ^^  +  ^  +  ^°^/2  ^^J"^     ro(^2  +^)  ~   ^o( 

Such  an  expression  in  rectangular  co-ordinates  as, 


represents  not  only  the  current  strength  but  also  its  phase. 

It  means,  in  Fig.  32,  that 
the  total  current  Of  has  the  two 
rectangular  components,  the  hor- 
izontal component  /  cos  <*>  =  /, 
and  the  vertical  component 
/  sin  o>  =  ir. 

Thus, 


tan  <o  =  -, 
i 


32- 


that  is,  the  tangent  function  of  the  phase  angle  is  the  verti- 
cal component  divided  by  the  horizontal  component,  or  the 
term  with  prefix/  divided  by  the  term  without/. 
The  total  current  intensity  is  obviously, 


The  capital  letter  /in  the  symbolic  expression  I  —  i  +j  ir 
thus  represents  more  than  the  /  used  in  the  preceding  for 
total  current,  etc.,  and  gives  not  only  the  intensity  but  also 
the  phase.  It  is  thus  necessary  to  distinguish  by  the  type 
of  the  latter,  the  capital  letters  denoting  the  resultant  cur- 


84  ELECTRICAL   ENGINEERING 

rent   in   symbolic  expression  (that  is,  giving  intensity  and 
phase)  from  the  capital  letters  giving  merely  the  intensity 
regardless  of  phase. 
That  is, 

/=/+.//' 

denotes  a  current  of  intensity 


and  phase 

/' 

tan  w  =  —  • 
t 

In  the  following,  dotted  italics  will  be  used  for  the  sym- 
bolic expressions  and  plain  italics  for  the  absolute  values  of 
alternating  waves. 

In  the  same  way  z  =  Vr2  +  x*  is  denoted  in  symbolic 
representation  of  its  rectangular  components  by 

Z  =  r  —  jx. 

When  using  the  symbolic  expression  of  rectangular  co- 
ordinates it  is  necessary  ultimately  to  reduce  to  common 
expressions. 

Thus  in  the  above  discussed  transformer  the  symbolic 
expression  of  primary  impressed  E.M.F. 


+  r0(*/i+  X)  +  %{< 
means  that  the  primary  impressed  E.M.F.  has  the  intensity, 


and  the  phase, 

tan    >= 


0  (ai\  +  K) 


This  symbolism  of  rectangular  components  is  the  quick- 
est and  simplest  method  of  dealing  with  alternating  current 
phenomena,  and  is  in  many  more  complicated  cases  the  only 


RECTANGULAR    CO-ORDINATES.  85 

method  which  can  solve  the   problem  at   all,   and  thus  the 
reader  should  become  fully  familiar  with  this  method. 

EXAMPLES. 

(1).  In  a  20  K.W.  transformer  the  ratio  of  turns  is  20  : 1, 
and  100  volts  are  required  at  the  secondary  terminals  at  full 
load.  What  is  the  primary  current,  the  primary  impressed 
E.M.F.,  and  the  primary  lag, 

a)  At  noninductive  load,  Wl  =  0 ; 

b)  With  «1  =  600lag  in  the  external  secondary  circuit; 

c)  With  <«>!=  —60°  lead  in  the  external  secondary  circuit? 
The  exciting  current  is  /«,'=  .3  -K4/  amperes,  at  ^=2000 

volts  impressed,  or  rather,  primary  induced  E.M.F. 

The  primary  impedance,  Z0  =  2  —  5/  ohms. 

The  secondary  impedance,  Zx  =  .004  —  .Ol/  ohms. 

We  have,  in  symbolic  expression,  choosing  the  secondary 
current  1^  as  real  axis  (see  page  86). 

(2).  eQ=  2000  volts  are  impressed  upon  the  primary  cir- 
cuit of  a  transformer  of  ratio  of  turns  20  : 1.  The  primary 
Impedance  is,  ZQ  =  2  -  5/,  the  secondary  impedance,  Zx= 
.004  —  .Oly,  and  the  exciting  current  at  /  =  2000  volts 
induced  E.M.F.  is  fm  =  .3  +  Aj,  thus  the  "primary  admit- 
tance," Y  =  &  =  (.15  +  2;)  10-3. 

What  is  the  secondary  current  and  secondary  terminal 
voltage,  and  the  primary  current,  if  the  secondary  circuit  is 
•closed  by, 

a),    resistance, 

Z  =  r  =  .5  —  noninductive  circuit. 
b\   impedance, 

Z  —  r  —  jx  =  .3  —  AJ  —  inductive  circuit. 
c).    impedance, 

Z  =  r  —  jx  =  .3  -f-  Aj  —  anti-inductive  circuit. 
Let, 

e  =  secondary  induced  E.M.F., 

assumed  as  real  axis  in  symbolic  expression  (see  page  86.) 


86 


ELECTRICAL   ENGINEERING. 


Is-  "*        2-, 

?53^S 

*    CJ  CM  c7 

+       +    1 


ill 

>> 


^ 

l~- 

co  > 


^ogg-og 


0  r^  ^0  0 

^  co  ^^  oo 

^  0  '^-N  CO 


KE  C  7'A  ArG  ULA  R    CO-  OKDINA  TES. 


87 


o  '<v.. 


. 

SIS 


.,.  _i_ 


£i>£l 
T._ja||  8 

co  +  po  o  U- 


a> 


1 


•N    t®.     -  JO  W  ^  S    ij  M  °° 

ls€kss  +s±§+ 

SN  S"3    2 

C^  r-"  «O 


Vi 

(N    1 


1 


_      I 

+  O  4-  5i 

r_(    I    i^  ' —  — 

co^^2,^^^c^  ^> 

^    ^O^T 


«ll   agig+S 

iCMCi          "»-|^QC)^- 


** 


§J     3 

J5     <•> 


<u   3   O 
fa   °   > 

^ 


W 


«     ty- 
bO    7* 


0) 
CJ 

G      •-; 


c  c  <u  cSrtrt-rtS.-r^ 

o  o  ^  g  g  g  3  8  1 3    *: 

oj  v  ^  •£,  'C  "C  X  -C   X  J3       VH         i-.     .S3 

C/3  C/2  O  0-  OH  PL,  H  OH  W  H      O        O      Ol  C/2 


.    . 

s  .11 


i  §111 

' 


C    " 


88  ELECTRICAL   ENGINEERING. 

3).  A  transmission  line  of  impedance  Z  —  r  —  j  x  — 
20  —  50 j  feeds  a  noninductive  receiving  circuit.  At  the 
receiving  end,  an  apparatus  is  connected  which  produces 
wattless  lagging  or  leading  currents  at  will  (synchronous 
machine)  .12000  volts  are  impressed  upon  the  line.  How 
much  lagging  and  leading  currents  respectively  must  be 
produced  at  the  receiving  end  of  the  line,  to  get  10,000 

volts, 

a),   at  no  load, 

b).    at  50  amperes  energy  current  as  load, 
c).    at  100  amperes  energy  current  as  load  ? 
Let 

e  =  10,000  =  E.M.F.  received  at  end  of  line. 
t\  =  energy  current,  /2  =  wattless  lagging  current ; 
thus, 

/=  1\  +/4  =  total  current  in  line. 

The  voltage  at  generator  end  of  line  is  then, 
£0=:  e  +  ZI 


=  (e  4-  n\ 

=  (10,000  +  20  /;  +  50  4)  +  /(20  /2  -  50  /;), 
or,  reduced, 

thus,  since  EQ  =  12,000, 

12,000  = 
a).    No  load, 

/i  =  0  ; 
thus, 


12,000  =  V(10,000  +  50  /2)2  4-  400  *22; 

hence, 

?2  =  +39.5  amps.,  wattless  lagging  current,  /=  4-39.5y. 

b).    Half  load, 
^  =  50; 


LOAD    CHARACTERISTIC    OF   TRANSMISSION  LINE.    89 

thus, 

12,000  =  V(ll,000  +  50/2y2-f-  (20/2-  2,500)2  ; 

hence, 

/2  =  +16  amps.,  lagging  current,  /=  50  +  16/ 

c).    Full  load, 


thus, 


12,000  =  V(12,000  -f  50  /2)2  +  (20  /2  -  5000)2  ; 

hence, 

4  =  —  27.13  amps.,  leading  current,  7=  100  —  27.13/. 

15.    LOAD  CHARACTERISTIC   OF   TRANSMISSION 

LINE. 

The  load  characteristic  of  a  transmission  line  is  the  curve 
of  volts  and  watts  at  the  receiving  end  of  the  line  as  func- 
tion of  the  amperes,  and  at  constant  E.M.F.  impressed  upon 
the  generator  end  of  the  line. 

Let  r  =  resistance,  x  —  reactance  of  the  line.  Its  impe- 
dance  z=  Vr2+^2  can  be  denoted  symbolically  by 

Z  =  r  —  jx. 

Let  EQ  =  E.M.F.  impressed  upon  the  line. 

Choosing  the  E.M.F.  at  the  end  of  the  line  as  horizontal 
component  in  the  polar  diagram,  it  can  be  denoted  by  E  =  e. 

At  noninductive  load  the  current  flowing  over  the  line  is 
in  phase  with  the  E.M.F.  e,  thus  denoted  by  I—i. 

The  E.M.F.  consumed  by  the  line  impedance  Z  =r—jx  is, 

E,=  ZI=(r-jx)i 
=  ri  —  jxi. 

Thus  the  impressed  voltage, 

EQ  =  E  -f  fi  =  e  +  ri  -  jxi. 
or,  reduced, 

JS0  =  V(>  +  ri^ 
and 


e  = 


E.M.F. 


90  ELECTRICAL   ENGINEERING. 


the  power  received  at  end  of  the  line. 

The  curve  of  E.M.F.  e  is  an  arc  of  an  ellipse. 
At  open  circuit  it  is, 

/  =  o, 

thus, 

e  =  EQ, 

/>=<>, 
as  to  be  expected. 

At  short  circuit, 

e  =  0, 

0  =  V^o2  -  X-*2  -  ri, 
thus, 

£        =  -°, 
2       *  ' 


that  is,  the  maximum  current  which  can  flow  over  the  line 
with  a  noninductive  receiver  circuit  and  at  negligible  line 
capacity. 

The  condition  of  maximum   power  delivered  over  the 

line  is  jp 

~  =  0, 

di 

that  is, 


substituted, 
and  expanded, 


hence, 

e  =  27, 


-  =  r1  is  the  resistance  or  effective  resistance  of  the  re- 
ceiving circuit. 
That  is,  — 
Maximum  power  is  delivered  into  a  nonindctive  receiving 


LOAD    CHARACTERISTIC    OF    TRANSMISSION  LINE.     91 

circuit  over  an  inductive  line  impressed  by  constant  E.M.F., 
if  the  resistance  of  the  receiving  circuit  equals  the  impe- 
dance of  the  line. 

^  =  z. 

In  this  case  the  total  impedance  of  the  system  is, 

Z0  =  Z  +  rl  =  r  +  z  —  jx, 
or 


z0  =  V(r  +  zf 
thus  the  current, 


The  power  transmitted  is, 
P  _  /  ir  


(r 


that  is,  — 

The  maximum  power  which  can  be  transmitted  over  a 
line  of  resistance  r  and  reactance  x  is  the  square  of  the  im- 
pressed E.M.F.  divided  by  twice  the  sum  of  resistance  and 
impedance  of  the  line. 

At  x  =  0,  this  gives  the  common  formula, 


47 


INDUCTIVE    LOAD. 

With  an  inductive  receiving  circuit  of  lag  angle  w,  or 
power  factor  /=cos  w,  and  inductance  factor  ^  =  sin  w,  at 
E.M.F.  E=e  at  receiving  circuit,  the  current  is  denoted 

by, 


thus  the  E.M.F.  consumed  by  the  line  impedance  Z=r~jx  is, 

&;-£/~/(/fj 

=  I[(rp  -f- 


92  ELECTRICAL   ENGINEERING. 

and  the  generator  voltage, 


=  \e  +  /(r/  +  *£)]  +//fa  -  */), 
or,  reduced, 


and, 


the  power  received  is  the    E.M.F.   times  the  energy  com- 
ponent of  current,  thus, 


-  /«/(#+**), 

the  curve  of  E.M.F.  e  as  function  of  the  current  /  is  again 
an  arc  of  an  ellipse. 

At  short  circuit  e  =  0,  thus,  substituted 

7"  "^^0 

Z 

the  same  value  as  at  noninductive  load,  as  is  obvious. 

The    condition  of  maximum  output  delivered  over  the 
line  is, 

that  is,  differentiated  and 


02  -  /2  (rq  -  x£?  =  e  +  I(rp 

substituted,  and  expanded, 

<?  =  72  (^  +  x*) 
=  7V; 

««A; 
or, 


#!  =  -  is  the  impedance  of  the  receiving  circuit. 

That  is,  the  power  received  in  an  inductive  circuit  over 
an  inductive  line  is  a  maximum,  if  the  impedance  of  the 
receiving  circuit,  zv  equals  the  impedance  of  the  line,  z. 


LOAD    CHARACTERISTIC    OF   TRANSMISSION-  LINE.     93 

In  this  case  the  impedance  of  the  receiving  circuit  is, 

Zi=z(p-jg)t 
and  the  total  impedance  of  the  system, 

ZQ=Z  +  Z, 

=  r—jx  +  z(p  -jq] 
=  (r+pz)-j(x  +  qz}. 

Thus  the  current, 


and  the  power, 


qzf 


EXAMPLES. 

(1.)  12000  volts  are  impressed  upon  a  transmission  line 
of  impedance  Z=r  —  jx  =  20  —  50/.  How  does  the  voltage 
and  the  output  in  the  receiving  circuit  vary  with  the  current, 
at  noninductive  load  ? 

Let  e  =  voltage  at  receiving  end  of  line,  i  =  current,  thus 
=  ei  =  power  received.     The  voltage  impressed  upon  the 
line  is  then, 

EQ  =  e  +  zi 

=  e  -f-  ri—jxi; 
or,  reduced, 


Since  £0  =  12000, 


12,000  =  V(V  +  r/)2  +  ^/2  =  V(^  +  20/)2  +  2,500  / 


<?  =  V12,0002  -  ^72  -  r/  =  Vl2,0002  -  2,500  /2- 
The  maximum  current  is  at  e  =  Q, 

0  =  Vl2,0002  -  2,500  i*  -  20  /, 
thus, 

/  =  223. 


94  ELECTRICAL   ENGINEERING. 

Substituting  for  it  gives  the  values  plotted  in  Fig.  33. 


Volts 
11000 


K.W. 
1000 


COJ 


9000 


\ 


8000 
7000 


0000 
JiOCO 
4000 
3000 


\ 


2000 


V 


Amps. 


40         00          80        100        120        no        1GO       ISO         200       220 
Fig.  33. 


i. 

e. 

/  =  ei. 

0 

12,000 

0 

20 

11,500 

230  x  10' 

40 

11,000 

440  x  103 

60 

10,400 

624  x  103 

80 

9,700 

776  x  103 

100 

8900 

890  x  103 

120 

8,  COO 

963  x  103 

140 

6,040 

971  x  103 

160 

5,750 

920  x  103 

180 

4,240 

764  x  103 

200 

2,630 

526  x  103 

220 

400 

8J  x  103 

223 

0 

0 

16.    PHASE   CONTROL  OF    TRANSMISSION    LINES. 

If 'in  the  receiving  circuit  of  an  inductive  transmission  line 
the  phase  relation  can  be  changed,  the  drop  of  voltage  in 
the  line  can  be  maintained  constant  at  varying  loads  or  even 


2'HASE    CONTROL    OF   TRANSMISSION  LINES.  95 

decreased  with  increasing  load  ;  that  is,  at  constant  generator 
voltage  the  transmission  can  be  compounded  ior  constant 
voltage  at  the  receiving  end,  or  even  over-compounded  for 
a  voltage  increasing  with  the  load. 

1.     COMPOUNDING    OF    TRANSMISSION    FOR    CONSTANT 
VOLTAGE. 

Let  ;-=  resistance,  x-=  reactance  of  transmission  line, 
eQ  =  voltage  impressed  upon  the  beginning  of  the  line,  e  = 
voltage  received  at  end  of  line. 

Let  i  =  energy  current  in  the  receiving  circuit;  that  is, 
P  —  ei  —  transmitted  power,  and  /i=  watt  less  current  pro- 
duced in  the  system  for  controlling  the  voltage.  z\  shall  be 
considered  positive  as  lagging,  negative  as  leading  current. 

The  total  current  is  then,  in  symbolic  representation, 

/=  /+//i; 
the  line  impedance, 

Z  =  r—jx; 

thus  the  E.M.F.  consumed  by  the  line  impedance, 


=  ri  +  jn\  —  jxi  —J2xt\  ; 
and  substituting 

X—  i. 

fi  =  (ri  +  *0  +J  ("i  —  x*')' 
Hence  the  voltage  impressed  uppn  the  line 


=  (e  +  ri  +  xt\)  +j(rt\  - 
or  reduced, 


CQ  =      (>  -h  ri  +  xij  -h  (n\  -  xi)\ 

If  in  this  equation  e  and  e0  are  constant,  iv  the  wattless 
component  of  current,  is  given  as  function  of  the  energy 
current  i  and  thus  of  the  load  ei. 

Hence  either  e0  and  e  can  be  chosen,  or.  one  of  the 
E.M.F.'s  eQ  or  e  and  the  wattless  current  t\  corresponding 
to  a  given  energy  current  i. 


96  ELECTRICAL   ENGINEERING. 

If  t\  =  0  at  2=0,  and  c  is  assumed  as  given, 
Thus, 


e  =       <?  +  r  +  x       +  (r    -  xi  ), 
2  ^  (r/  +  */\)  +  (r2  +  *2)  (/»  -  /i2)  =  0. 


As  seen,  in  this  equation  t\  must  always  be  negative,  that 
is,  the  current  leading. 

From  this  equation  it  follows, 


ex  ± 


Thus  the  wattless  current  iv  must  be  varied  by  this  equation 
to  maintain  constant  voltage  e  =  e0  irrespective  of  the  load  ci. 
/i  becomes  impossible  if  the  term  under  the  square  root 
becomes  negative,  that  is,  at  the  value, 


or, 


At  this  point  the  power  transmitted  is 


This  is  the  maximum  power,  which  can  be  transmitted 
without  drop  of  voltage  in  the  line,  at  an  energy  current  i  = 


The  wattless  current  corresponding  hereto  is,  since  the 
square  root  becomes  zero, 

ex 

/i=  -^ 

thus  the  ratio  of  wattless  to  energy  current,  or  the  tan.  of. 
the  phase  angle  of  the  receiving  circuit,  is 

/i  x 

tan  a    =  —  =  ---  • 


A  larger  amount  of  power  is  transmitted  if  e0  is  chosen 
,  a  smaller  amount  of  power  if  e0<  e. 


PHASE    CONTROL    OF   TRANSMISSION  LINES.  97 

In  the  latter  case  ^  is  always  leading;  in  the  former  case 
^  is  lagging  at  no  load,  becomes  zero  at  some  intermediate 
load,  and  leading  at  higher  load. 

If  the  line  impedance  Z=r—jx  and  the  received  voltage 
e  is  given,  and  the  energy  current  iQ  at  which  the  wattless 
current  shall  be  zero,  the  voltage  at  the  generator  end  of  the 
line  is  determined  hereby  from  the  equation, 


*0  =  V(<?  +  ri  +  xt'tf  +  (rt\  -  xif, 
by  substituting 

i\  =  0,  /  =  4 


as 


'o  =  V(*  +  r/0)2+ 
Substituting  this  value  in  the  general  equation, 


<?o=  V(<?  +  ri  +  xi^f  +  (ri  -  xi 
gives 

(e  +  ^0)2  +  x*tf  =(e  +  ri  +  ^)2  +  ( 
as  equation  between  /  and  iv 

If  at  constant  generator  voltage  eQ, 
at  no  load, 

and  at  the  load, 

it  is,  substituted,  — 
no  load, 


load 


<?o  =  V(<r0  +  rttf  +  r\\ 
Thus, 

(<?0  -f  xqf  +  a-y  •=  (<?0  +  r/o)2  -f  rj£\ 
or,  expanded, 

This  equation  gives  q  as  function  of  z'0,  e0,  r,  x. 

If  now  the  wattless  current  t\  varies  as  linear  function  of 
the  energy  current  /,  as  in  case  of  compounding  by  rotary 
converter  with  shunt  and  series  field,  it  is, 


98  ELECTRICA.L    ENGINEERING. 

Substituting  this  value  in  the  general  equation, 
(<?0  +  r/0)2  H-  *2/02  =  (e  +  ri  +  aVJ->  (n\  -  a-/)2 

gives  e  as  function  of  i,  that  is,  gives  the  voltage  at  the  re- 
ceiving end  as  function  of  the  load,  at  constant  voltage  <?0  at 
the  generating  end,  and  e==c0  for  no  load, 

i  =  0,    /i  =  q, 
and  ^  =  e0  for  the  load, 

*  =  4>   >i  =  0. 


Between  /'=  0  and  z  =  /0,  ^>^0,  and  the  current  is  lagging. 

Above  i  =  /0,  e  <  e9  and  the  current  is  leading. 

By  the  reaction  of  the  variation  of  e  from  e0  upon  the 
receiving  apparatus  producing  wattless  current  iv  and  by 
magnetic  saturation  in  the  receiving  apparatus,  the  deviation 
o.  e  from  eQ  is  reduced,  that  is,  the  regulation  improved. 

2.      OVER-COMPOUNDING    OF    TRANSMISSION    LINES. 

The  impressed  voltage  at  the  generator  end  of  the  line 
was  found  in  the  preceding, 


+  (H^  -  xt*. 

If  the  voltage  at  the  end  of  the  line  e  shall  rise  propor- 
tionally to  the  energy  current  z,  then 


<?  =  /?!  +  at. 
Thus 


e0  =      h  -f  (a  +  r)  i  +  xitf  +  (ri,  -  xt)\ 

And  herefrom  in  the  same  way  as  in  the  preceding  we 
get  the  characteristic  curve  of  the  transmission. 

If  e0=ev  ^  =  0  at  no  load,  and  is  leading  at  load.  If 
e0  <  ev  t\  is  always  leading,  and  the  maximum  output  less  than 
before. 

If  e0  >  ev  ^  is  lagging  at  no  load,  becomes  zero  at  some 
intermediate  load,  and  leading  at  higher  load.  The  maxi- 
mum output  is  greater  than  at  e0  =  rr 


E   CONTROL    OF   TRANSMISSION  LINES.  99 

The  greater  a,  the  less  is  the  maximum  output  at  the 
same  CQ  and  er 

The  greater  ev  the  greater  is  the  maximum  output  at  the 
same  el  and  a,  but  the  greater  at  the  same  time  the  lagging 
current  (or  less  the  leading  current)  at  no  load. 

EXAMPLES. 

(1.)  A  constant  voltage  of  e0  is  impressed  upon  a  trans- 
mission line  of  impedance  Z  =  r  —jx  =  10  —  20;.  The  voltage 
at  the  receiving  end  shall  be  10000  at  no  load  as  well  as  at 
full  load  of  75  amps,  energy  current.  The  wattless  current 
in  the  receiving  circuit  is  raised  proportionally  to  the  load,  so 
as  to  be  lagging  at  no  load,  zero  at  full  load  or  75  amps.,  and 
leading  beyond  this.  What  voltage  e0  has  to  be  impressed 
upon  the  line,  and  what  is  the  voltage  e  at  the  receiving  end 
at  i,  §-,  and  \\  load  ? 

Let  /=*i+./y2  =  current,  E=e  voltage  in  receiving  circuit. 
The  generator  voltage  is  then, 


=  (e  +  ;-/;  +  .r/o)  -f  y(r/2  —  xt\) 
=  (e  +  10  /i  +  20  4)  +/  (10  /2  -  20  /i),  . 
cr,  reduced, 

e*  =(*  +  rt\  +  */2)2  +  (ri2  -  xtf 
or, 

=  (e  +  10  /;  +  20  4)2  +  (10  4  -  20  1\)*. 
At 

i\  =  75,    ^2  =  0,    e  =  10,000, 

thus,  substituted, 

t*  =  10,7502  +  1,5002  =  117.81  x  106; 
hence, 

CQ  =  10,860  volts  is  the  generator  voltage. 
At 

t\  =  0,  e  =  10.000,   e(}  =  10,860,  let  ?2  = 

these  values  substituted  give, 

117.81  x  106  =  (10,000  +  20  qj  +  100  f 
=  100  x  106  +  400^  x  103  •+ 


100 
or, 


ELECTRICAL   ENGINEERING. 


q  =  44.525  -  1.25  f  10~s ; 

this  equation    is  best    solved    by  approximation,   and   then 
gives, 

q  =  42.3  amps,  wattless  lagging  current  at  no  load. 

Since, 

^2  =  0?  +  «i 

it  follows, 


or, 


e  =  V117.81  X  106  -  (10  /2  -  20  itf  -  (10  ^  +  20  /2). 
Substituting  herein  the  values  of  z\  and  z'2  gives  *?. 


*v 

i.,. 

e. 

0 

42.3 

10,000 

25 

28.2 

10,038 

50 

14.1 

10,038 

75 

0 

10,000 

100 

-14.1 

9,922 

125 

-28.2 

9,803 

(2.)  A  constant  voltage  e0  is  impressed  upon  a  trans- 
mission line  of  impedance  Z=  r  —  jx=  10  —  10y.  The  voltage 
at  the  receiving  end  shall  be  10,000  at  no  load  as  well  as  at 
full  load  of  100  amps,  energy  current.  At  full  load  the  total 
current  shall  be  in  phase  with  the  E.M.F.  at  the  receiving 
end,  and  at  no  load  a  lagging  current  of  50  amps,  is  per- 
mitted. How  much  additional  reactance»^0  is  to  be  inserted, 
what  must  be  the  generator  voltage  e0,  and  what  will  be  the 
voltage  e  at  the  receiving  end  at  \  load  and  at  H  l°ad,  if  the 
wattless  current  varies  proportionally  to  the  load  ? 


Let  Xfi=  additional  reactance  inserted  in  circuit. 
Let      =  /  +      =  current. 


Then, 


ftf  +  10  iz  -  x£f, 


PHASE    COATTROL    OF  TRANSMISSION  LINES.         101 


where, 

xl  =  x  -+-  xQ  =  total  reactance  of  circuit  between  e  and  ^0. 

At  no  load, 

t\  =  0,   /2  =  50,   e  =  10,000, 
thus,  substituted, 

e*  =  (10,000  +  50*!)*  +  250,000. 
At  full  load, 

/i  =  100,    4  =  0,    ^  =  10,000; 
thus,  substituted, 

<-o2  =  121  x!06  + 10,000  x*. 

Combined,  these  give, 

(10,000  +  50  ^)2  +  250,000  =  121  x  106  +  10,000  jtf; 
hence, 

Xl  =  66.5  ±  40.8 

=  107.3    or   25.7  ; 
thus, 

x0  =  x1  —  x  =  97.3,  or  15.7  ohms  additional  reactance. 

Substituting, 

*x  =  25.7, 
gives, 

,0*=  (e  +  10  i,  +  25.7 ty  +  (10  /2  -  25.7  z')2 ; 

but,  at  full  load, 

t\  =  100,   4  =  0,   <?  =  10,000,  - 
which  values  substituted  give, 

e*  =  121  x  106  +  6.605  x  106  =  127.605  x  106, 
e0  =  11,300  generator  voltage. 
Since 


it  follows 


e  =  V^02-  (10/2-  25.7 ty  -  (10 /!  +  25.74), 


e  =  V127.605  X 106-  (10/2-  25.7/;)2  -  (10 ^  +  25.7/2). 
Substituting  for  ^  and  /2,  gives  e. 


z\. 

4- 

e. 

0 

50 

10,000 

50 

25 

10,105 

100 

0 

10,000 

150 

-25 

9,658 

102  ELECTRICAL    ENGINEERING. 

(3)  In  a  circuit  whose  voltage  e0  fluctuates  by  20%  be- 
tween 1800  and  2200  volts,  a  synchronous  motor  of  internal 
impedance  Z0  =  r0  —jxQ  =  .5  —  5/  is  connected  through  a  re- 
active coil  of  impedance'  Z^  —  r^  —jx^  =  .5  —  10/,  and  run  light, 
as  compensator  (that  is,  generator  of  wattless  currents). 
How  will  the  voltage  at  the  synchronous  motor  terminals 
cv  at  constant  excitation,  that  is,  constant  counter  E.M.F. 
^  =  2000,  vary  as  function  of  c^  at  no  load,  and  at  a  load  of 
z  =100  amps,  energy  current,  and  what  will  be  the  wattless 
current  in  the  synchronous  motor  ? 

Let  /=  t\  -h/V2=  current  in  receiving  circuit  of  voltage 
er  Of  this  current  /,  jiy  flows  into  the  synchronous  motor 
of  counter  E.M.F.  ey  and  thus, 


=  e 
or,  reduced, 


In  the  supply  circuit  the  voltage  is, 


or,  reduced, 


Substituting  in  the  equations  for  e^  and  e*>  the  above  values 
of  r0  and  ^  : 
At  no  load 

/i  =  0, 
we  have 


At  full  load, 

/V=  100, 

^=(>  +  54)2  +  .25/22 
^  =  (^  +  50  +  15  /2)2  +  (4  -  1,000)2, 
^  =  2.000  substituted, 


IMPEDANCE  AND   ADMITTANCE. 


103 


at  no  load, 


gives, 


at  full  load, 


e*  =  (2,050  +15  ty  +  (/2  -  1,000)2. 

Substituting  herein  e0  =  successively  1,800,  1,900,  2,000, 
2,100,  2,200,  gives  values  of  z'2>  which,  substituted  in  the 
equation  for  e{,  give  the  corresponding  values  of  ev  as  re- 
corded in  the  following  table. 

As  seen,  in  the  local  circuit  controlled  by  the  syn- 
chronous compensator,  and  separated  by  reactance  from  the 
main  circuit  of  fluctuating  voltage,  the  fluctuations  of  volt- 
age appear  in  a  greatly  reduced  magnitude  only,  and  can  be 
entirely  eliminated  by  varying  the  excitation  of  the  syn- 
chronous compensator. 


e  —  2,0000. 

* 

No.  LOAD, 

t\  —  0, 

FULL  LOAD, 

t\  =  100, 

1,800 

-  13.3 

1,037 

-39 

1,810 

1,900 

-    7.C8 

1,965 

-30.1 

1,850 

2,000 

0 

2,OCO 

-22 

1,865 

2,100 

4-    6.7 

2,035 

-13.5 

1,935 

2,200 

+  13.3 

2,074 

-   6.5 

1,970 

17.    IMPEDANCE  AND    ADMITTANCE. 

In   direct   current   circuits   the   most    important   law   is 
Ohm's  law  : 


or, 
or, 


<?=  tr, 


104  ELECTRICAL   ENGINEERING. 

where  e  is  the  E.M.F.  impressed  upon  resistance  r  to  pro- 
duce current  i  therein. 

Since  in  alternating  current  circuits  by  the  passage  of  a 
current  i  through  a  resistances  additional  E.M.F.'s  may  be 

produced  therein,  when  applying  Ohm's  law,  i  =  -  to  alter- 

nating current  circuits,  e  is  the  total  E.M.F.  resulting  from 
the  impressed  E.M.F.,  and  all  E.M.F.'s  produced  by  the  cur- 
rent i  in  the  circuit. 

Such  counter  E.M.F.'s  may  be  due  to  inductance,  as  self 
or  mutual  inductance,  to  capacity,  chemical  polarization,  etc. 

The  counter  E.M.F.  of  self-induction,  or  E.M.F.  induced 
by  the  magnetic  field  produced  by  the  alternating  current  i, 
is  represented  by  a  quantity  of  the  same  dimension  as  resist- 
ance, and  measured  in  Ohms  :  reactance  x.  The  E.M.F. 
consumed  by  reactance  x  is  in  quadrature  with  the  current, 
that  consumed  by  resistance  r  in  phase  with  the  current. 

Reactance  and  resistance  combined  give  the  impedance, 


or  in  symbolic  or  vector  representation, 
Z  =  r  —  jx. 

In  general  in  an  alternating  current  circuit  of  current  z, 
the  E.M.F.  e  can  be  resolved  in  two  components,  an  energy 
component  ^  in  phase  with  the  current,  and  a  wattless  com- 
ponent ^2  in  quadrature  with  the  current. 

The  quantity 

el      energy  E.M.F.  or  E.M.F.  in  phase  with  the  current  _ 
i  current 

is  called  the  effective  resistance. 

The  quantity 

*?2  _  wattless  E.M.F.  of  E.M.F.  in  quadrature  with  the  current  _ 
/  current 

is  called  the  effective  reactance  of  the  circuit. 

And  the  quantity, 


IMPEDANCE   AND   ADMITTANCE.  105 

or  in  symbolic  representation, 

is  the  impedance  of  the  circuit. 

If  power  is  consumed  in  the  circuit  only  by  the  ohmic 
resistance  r,  and  counter  E.M.F.  produced  only  by  self- 
induction,  the  effective  resistance  1\  is  the  true  or  ohmic 
resistance  r,  and  the  effective  reactance  x^  is  the  true  or  self- 
inductive  reactance  x. 

By  means  of  the  terms,  effective  resistance,  effective 
reactance,  and  impedance,  Ohm's  Law  can  be  expressed  in 
alternating  circuits  in  the  form, 

e 


or,  z1  = 

or  in  symbolic  or  vector  representation 


or, 
or, 


In  this  latter  form  Ohm's  Law   expresses  not  only  the 
intensity  but  also  the  phase  relation  of  the  quantities. 
It  is, 

e\  —  ir\  =  energy  component  of  E.M.F., 
<?2  =  ix^  =  wattless  component  of  E.M.F. 

Instead  of  the  term   impedance  z—^  with    its   components 
the  resistance  and  reactance,  its  reciprocal  can  be  introduced 

i      1 

e      z* 

which  is  called  the  admittance. 

The  components  of  the  admittance  are  called  the  conduc- 
tance and  susceptance. 


106  ELECTRICAL   ENGINEERING. 

Resolving  the  current  i  into  an  energy  component  t\  in 
phase  with  the  E.M.F.  and  a  wattless  component  z'2  in  quad- 
rature with  the  E.M.F.,  the  quantity 

t\       energy  current,  or  current  in  phase  with  E.M.F., 
~e  =  E.M.F. 

is  called  the  conductance. 
The  quantity, 

4  _  wattless  current,  or  current  in  quadrature  with  E.M.F.,  _ 
7=  E.M.F. 

is  called  the  susceptance  of  the  circuit. 

The  conductance  represents  the  current  in  phase  with 
the  E.M.F.  or  energy  current,  the  susceptance  the  current 
in  quadrature  with  the  E.M.F.  or  wattless  current. 

Conductance  g  and  susceptance  b  combined  give  the 
admittance 

y  =  VFT^2, 

or  in  symbolic  or  vector  representation  : 

Y=g+jb. 
Ohm's  Law  can  thus  also  be  written  in  the  form  : 


•or, 
or, 


or,  in  symbolic  or  vector  representation, 

f=EY=E(g+jb)\ 

E-t  =  -J-', 

Y      g+jb' 


and,  j\  =  eg-  =  energy  component  of  current, 

iz  =  eb  =  wattless  component  of  current. 

According  to  circumstances,   sometimes  the  use  of  the. 
terms  impedance,  resistance,  reactance,  sometimes  the  use 


IMPEDANCE  AND   ADMITTANCE.  107 

of  the  terms  admittance,  conductance,  susceptance,  is  more 
convenient. 

Since,  in  a  number  of  series  connected  circuits,  the  total 
E.M.F.,  in  symbolic  representation,  is  the  sum  of  the  indi- 
vidual E.M.F.'s,  it  follows  : 

"  In  a  number  of  series-connected  circuits,  the  total  im- 
pedance, in  symbolic  expression,  is  the  sum  of  the  impedances 
of  the  individual  circuits  connected  in  series." 

Since  in  a  number  of  parallel  connected  circuits  the  total 
current,  in  symbolic  representation,  is  the  sum  of  the  indi- 
vidual currents,  it  follows  : 

"  In  a  number  of  parallel  connected  circuits,  the  total 
.admittance,  in  symbolic  expression,  is  the  sum  of  the  admit- 
tances of  the  individual  circuits  connected  in  parallel." 

Thus  in  series  connection  the  use  of  the  term  impedance, 
in  parallel  connection  the  use  of  the  term  admittance,  is 
generally  more  convenient. 

Since  in  symbolic  representation, 


or, 

that  is,  (r  -jx)  (g  +  jV)  =  1  ; 

It  follows  that, 

(rg  +  xb)  +  j(rb  -  xg)  =1; 
that  is,  rg  4-  xb  =  1, 

rb  —  xg  =  0. 

Thus,  g       =  g, 

f+P      f 
b  b 


108  ELECTRICAL   ENGINEERING. 

or  absolute, 


Thereby  the  admittance  with  its  components,  the  con- 
ductance and  susceptance,  can  be  calculated  from  the 
impedance  and  its  components,  the  resistance  and  reactance, 
and  inversely. 

If  .r  =  0,  z  =  randg-  =—  that  is,  g  is  the  reciprocal  of 

the  resistance  in  a  noninductive  circuit,  not  so,  however,  in 
an  inductive  circuit. 

EXAMPLES. 

(1.)  In  a  quarter-phase  induction  motor,  at  E.M.F.  e  = 
110  impressed  per  phase,  the  current  is,  f0  =  ?\  +/4  =  100 
+  100;'  at  standstill,  the  torque  =  T0. 

The  two  phases  are  connected  in  series  in  a  single-phase 
circuit  of  E.M.F.  e  =  220,  and  one  phase  shunted  by  a  con- 
denser of  1  ohm  capacity  reactance. 

What  is  the  starting  torque  T  of  the  motor  under  these 
conditions,  compared  with  Tw  the  torque  on  a  quarter-phase 
circuit,  and  what  the  relative  torque  per  volt-ampere  input,  if 
the  torque  is  proportional  to  the  product  of  the  E.M.F.'s 
impressed  upon  the  two  circuits  and  the  sine  of  the  angle 
of  phase  displacement  between  them  ? 

In  the  quarter-phase  motor  the  torque  is, 

T0  =  a*  =  12,100  a, 
where  a  is  a  constant.     The  volt-ampere  input  is, 

Co  =  2^  V?+tf  =  31,200; 

hence,  the  "  apparent  torque  efficiency,"  or  torque  per  volt- 
ampere  input, 

4  =§=-388, 


'  THE 

:.->    •         -• 


IMPEDANCE   AND   ADMITTANCE.  109 

The  admittance  per  motor  circuit  is, 


K=/=.91 


the  impedance, 

Z  =  -  =         11Q        =          110  (100  -  IQOy) 

/      100  H-  lOOy       (100  +  lOOy)  (100  -  100/)  "~        "  '55j' 

the  admittance  of  the  condenser  : 


thus,  the  joint  admittance  of  the  circuit  shunted  by  the  con- 
denser, 


=  .91  -  .09y; 
thus,  its  impedance, 


==  1<09 


l      Fx      .91  -  .09y      .912+  .092 
and  the  total  impedance  or  the  two  circuits  in  series, 


=  1.64  -  .44y. 
Hence,  the  current,  at  impressed  E.M.F.  e  =220, 

7=  '+  ••  =  -!-     22°    _  220 (1.64  +  .44y) 

=  *1  -t-772    ^  -  j  64  _  44y  ~     1  g^  +  442 

=  125  +  33.5y; 
or,  reduced, 

1=  V1252  +  33.52 
=  129.4  amps. 

thus,  the  volt-ampere  input, 

Q  =  eS=220  x  129.4 

=  28,470. 

The  E.M.F. 's  acting  upon  the  two  motor  circuits  respec- 
tively are, 

JEl  =  7Z1=  (125  +  33.5s")  (1-09  +  .lly)  =  132.8  +  50.4/, 
and, 

E  =IZ  =  (125  +  33.5y)  (.55  -  .56/)  =  87.2  -  50.4y. 


110  ELECTRICAL   ENGINEERING. 


Thus,  their  phases  are  : 

50  4 

tan  ^  =  —         ~   =  —  -30  ;  hence,  Wl  =  —  21°  ; 


tan  u/  =  +    =^  =  +  .579  ;  hence,  w'  =  +  30°; 
and  the  phase  difference, 

/  p--f  O 

w   =  u>    —  a^  =  Ol  . 

The  absolute  values  of  these  E.M.F.'s  are, 


el  =  V132.8  +  50.42  =  141.5, 


/  =  V87.22  -  50.42  =  100.7  ; 
thus,  the  torque, 

T  '  =  ae^e'  sin  w 
=  11,100*; 

thus,  the  apparent  torque  efficiency, 

T  11,100  a 
/=-  = 


Hence  it   is,  compared  with   the  quarter-phase    motor, 
Relative  torque, 

7-11,100* 

r0     12,100* 

Relative  torque  per  volt-ampere,  or  relative  apparent  torque 
efficiency, 

'        *9a=  1.005. 


/0      .388  a 

(2).  At  constant  field  excitation,  corresponding  to  a 
nominal  induced  E.M.F.  e0  =12,000,  a  generator  of  synchro- 
nous impedance  Z0=  r0  —  j x^  =.6  —  60/  feeds  over  a  transmis- 
sion line  of  impedance  Z^  =  1\  —jx^  =12-18;',  and  of  capacity 
susceptance  .003,  a  noninductive  receiving  circuit.  How 
will  the  voltage  at  the  receiving  end,  e,  and  the  voltage  at 
the  generator  terminals  ev  vary  with  the  load,  if  the  line 
capacity  is  represented  by  a  condenser  shunted  across  the 
middle  of  the  line  ? 


IMPEDANCE  AND   ADMITTANCE.  Ill 

Let  /=  i  =  current  in  receiving  circuit,  in  phase  with  the 
E.M.F.,  £  =  e. 

The  voltage  in  the  middle  of  the  line  is, 


=  e  +  6  /  -  9  if. 

The  capacity  susceptance  of  the  line  is,  in  symbolic  ex- 
pression,  Y=   —  .003;',  thus  the  charging  current, 

/2  =  £2  Y  =  -  .003  j(e  +  6  /  -  9  /» 

=  -.027i  -/(.003*  +  .018*'), 
and  the  total  current, 

£  =  /  +  72  =  .973  /  -j  (.003  e  +  .018  /). 
Thus,  the  voltage  at  the  generator  end  of  the  line, 


—  9/)  [.973/—  y(. 
=  (.973^  +  11.68?)  -/  (17.87  /  +.018  e), 

and  the  nominal  induced  E.M.F.  of  the  generator, 


=  (.973  e  +  11.68  /  )  -y  (17.87  /  +.018  <r)  +  (.6  -  60y) 

[.973/-y(.003^+  .0180] 
=  (.793  e  +  11.18  0  -j  (76.26  /  +  .02  *)  ; 

or,  reduced,  and  e0  =  12,000  substituted, 

144  x  106  =  (.793  e  +  11.18  ij  +  (76.26  /  +  .02  e?; 
thus, 

r  +  33  ei  +  9,450  18  =  229  x  106 


^  =  -  16.5.  /  +  V229  x  10C  -  9,178  i\ 
and, 


et=  V(.973  e  +  11.68  /)2  +  (17.87  /  +  .018  ef, 
at, 

/  =  0,    ^  =  15,133,   fl=  14.700; 
at, 

e=  Q:    i=ir:5G,      r=3:-527. 


112  ELECTRICAL   ENGINEERING. 

Substituting  different  values  for  i,  gives, 


/. 

e. 

it* 

i. 

e. 

e\- 

0 

15,133 

14,700 

100 

10,050 

11,100 

25 

14,488 

14,400 

125 

7,188 

8,800 

50 

13,525 

13,800 

150 

2,325 

4,840 

75 

12,063 

12,730 

155.6 

0 

3,327 

which  values  are  plotted  in  Fig.  34. 


rolts 
15000 

••      • 

^=55 

^ 

^5- 

11000 

^ 

V^ 

\ 

13000 

Ope, 

iCiix 

uit 

x 

\ 

s, 

<;xx 

12000 

atUe 

nera 

or 

s. 

e 

v 

^> 
r^v 

**-, 

11000 

35 

^X 

>fe 

^^ 

10000 

V 

^X 

^ 

k 

9000 

*4 

\ 

8000 

\ 

\ 

7000 

\ 

\ 

6000 

\ 

\ 

5000 

v 

\ 

\ 

4000 

\ 

\ 

3000 

\ 

2000 

\ 

1000 

Ene 

^c 

!urre 

atR< 

c'd 

\ 

0 

10   20  30   10   50   CO   70   80   90   100  110  120  130  .110 
Fig.  34. 

18.    EQUIVALENT   SINE  WAVES. 

In  the  preceding  chapters,  alternating  waves  have  been 
assumed  and  considered  as  sine  waves. 

The  general  alternating  wave  is,  however,  never  com- 
pletely, frequently  not  even  approximately  a  sine  wave. 

A  sine  wave  having  the  same  effective  value,  that  is,  the 
same  square  root  of  mean  squares  of  instantaneous  values, 


EQUIVALENT  SINE    WAVES.  113 

as  a  general  alternating  wave,  is  called  its  corresponding 
"equivalent  sine  wave."  It  represents  the  same  effect  as 
the  general  wave. 

With  two  alternating  waves  of  different  shapes,  the  phase 
relation  or  angle  of  lag  is  indefinite.  Their  equivalent  sine 
waves,  however,  have  a  definite  phase  relation,  that  which 
gives  the  same  effect  as  the  general  wave,  that  is,  the  same 
mean  (ez). 

Hence  if  e=  E.M.F.  and  i=  current  of  a  general  alter- 
nating wave,  their  equivalent  sine  waves  are  defined  by, 


e0  =  Vmean  (V2), 

/0  =  Vmean  (V2); 
and  the  power  is, 

/0  =  e0i0  cos  e0i'0  =  mean  («"). 
Thus, 

mean  (et) 

COS  £/0  = —  • 

Vmean  (<?}  Vmean  (72) 

Since  by  definition  the  equivalent  sine  waves  of  the 
general  alternating  waves  have  the  same  effective  value  or 
intensity  and  the  same  power  or  effect,  it  follows,  that  in  re- 
gard to  intensity  and  effect  the  general  alternating  waves 
can  be  represented  by  their  equivalent  sine  waves. 

Considering  in  the  preceding  the  alternating  currents  as 
equivalent  sine  waves  representing  general  alternating  waves, 
the  investigation  becomes  applicable  to  any  alternating  cir- 
cuit irrespective  of  the  wave  shape. 

The  use  of  the  terms  reactance,  impedance,  etc.,  implies 
that  a  wave  is  a  sine  wave  or  represented  by  an  equivalent 
sine  wave. 

Practically  all  measuring  instruments  of  alternating  waves 
(with  exception  of  instantaneous  methods)  as  ammeters, 
voltmeters,  wattmeters,  etc.,  give  not  general  alternating 
waves,  but  their  corresponding  equivalent  sine  waves. 


114  ELECTRICAL   ENGINEERING. 

EXAMPLES. 

In  a  25-cycle  alternating  current  transformer,  at  1000 
volts  primary  impressed  E.M.F.,  of  a  wave  shape  as  shown 
in  Fig.  35  and  Table  I.,  the  number  of  primary  turns  is  500, 
the  length  of  the  magnetic  circuit  50  cm,  and  its  section 
shall  be  chosen  so  as  to  give  a  maximum  density  (B  =  15,000. 

At  this  density  the  hysteretic  cycle  is  as  shown  in  Fig. 
36  and  Table  II. 

What  is  the  shape  of  current  wave,  and  what  the  equiva- 
lent sine  waves  of  E.M.F.,  magnetism,  and  current? 

The  calculation  is  carried  out  in  attached  table. 


Fig.  35. 

In  column  (1)  are  given  the  degrees,  in  column  (2)  the 
relative  values  of  instantaneous  E.M.F.'s,  e  corresponding 
thereto,  as  taken  from  Fig.  35. 

Column  (3)  gives  the  squares  of  e.    Their  sum  is  24,939, 

24  939 
thus  the    mean  square,  - — *—-  =  1385.5,  and  the  effective 

lo 

value, 

/=  V138K5  =  37.22. 

Since  the  effective  value  of  impressed  E.M.F.  is=  1,000, 


E  Q  UI VA  LENT  SINE    WA  VES. 


115 


.  (8) 
FROM 
STERESI 
YCLE,  y. 


1- 


»ll 


s  « 


§O^ 
06 


»—  i'^ll>-rHO'?t*OC^D^ 

o  o  o  i-  §  ^5  -    o 


o  <  o  o  o  i- 


CO  CO  <>    i-l 


o 

(^  CO  ^ 


OiOOOOiO'OOO'-OO'OO 

§t^Oi—  lOt^i—  i  i—  lOCiCOOG<I 
Ct  Ci  t-  '-O  C^  r-l  i       i     rH  TH  7^  CO 


'r 

C5  O 


ooooooooooooooooo 

O  O'  O  O  >O  O  t^  O  00  l^  O  tT  O  O  O  O  O 

i  77  i  T  i  '  '  ' 


~ 

(?q  Oi  o  oo  i—  i  r-H  oo  oo  co  o  i-  co  ci  o  o  c?  I-H  <M  f^ 
OS  <M  «  O6  <O  O  >^  O  t-  O<N  tCO  ^  OOO  <N  CO  CO 


r-t  4-  "-1 


<><NCO'—  iO^^OO?OOCO(7^(7<IOOCD"*'^r1'^1 
rH  CO  t^  CO  i—  1  !M  O  O  O  r-t  t~  <N  CO  T-H  1C  O  O 


Ot^-r-iiOOiOOiOOOOOOOI^-QO 

r-it—  oroo-^ooco^Dooocot—  o 

(NCOOOOr-  ICOTtlOOOlOr-lt^COT-H 


rHCqCOCOCOCOCOrH 


ooooooooooooooooooo 
r-i<?qcoTt<»oot^cooiOrH(Mco^'Ooi-oo 


II       ^ 


0  0    .      00 

co  ao  co    •«« 


^-O      ^  Aj 

co  ie't^ 

rH       03  CO 


116 


ELECTRICAL   ENGINEERING 


the   instantaneous    values    are    en  = 


1,000 


column  (4). 


TABLE    II. 


/ 

a 

0 

±8 

,000 

2 

+  10, 

400- 

2,500 

4 

+  11, 

700  + 

5,800 

(3 

+  12, 

400  + 

9,300 

8 

+  13, 

0004- 

11,200 

10 

+  13, 

500  + 

12,400 

12 

+  13, 

900  + 

13,200 

14 

+  14, 

200  + 

13,800 

16 

+  14, 

500  + 

14,300 

18 

+  14, 

800  + 

14,700 

20 

+  15, 

000 

as  given  in 


f=Amp 


i  e  Turns  per  cm. 


Lengti 


JL   is   u 


36. 


These  values  added  give  column  (5),  the  integral  of  e# 


and  herefrom,   by  subtracting 


14,648 


=  7,324,  the  relative 


instantaneous  values  of  magnetic  induction  (B1,  in  column  (6). 


EQUIVALENT  SINE    WAVES, 


117 


Since  the  maximum  magnetic  induction  is  15,000  the 
instantaneous  values  are  (B  =  W-J—- ,  plotted  in  column  (7). 

I  ,0-Z-i 

From  the  hysteresis  cycle  in  Fig.  36  are  taken  the  values 
of  magnetizing  force  /,  corresponding  to  magnetic  induction 
<B.  They  are  recorded  in  column  (8),  and  in  column  (9)  the 


'Flux 


I 


Fig.  37. 


instantaneous     values    of    M.M.F.   $  =  lf,   where    /  =  50  = 
length  of  magnetic  circuit. 

gr 

/=-,  where  ;/  =  500  =  number  of  turns  of  the  electric 

circuit,  gives  thus  the  exciting  current  in  column  (10). 

Column  (11)  gives  the  squares  of  the  exciting  current, 

25  85 
**.    Their  sum  is  25.85,  thus  the  mean  square,      '       =1.436, 

lo 


118 


ELECTRICAL   ENGINEERING. 


and  the  effective  value  of  exciting  current,  i'  =  Vl.4^6  = 
1.198  amps. 

Column   (12)  gives  the  instantaneous  values  of  power, 

p  =  ie^    Their  sum  is  4,766,  thus  the  mean  power,  /'= 


=  264.8. 
Since, 


pr  =  e^i'  cos  o>, 


where  e'  and  i'  are  the  equivalent  sine  waves  of  E.M.F.  and 


Fig.  38. 


of  current  respectively,  and  o>  their  phase  displacement,  sub- 
stituting these  numerical  values  of  /',  /,  and  ir,  we  have 

264.8  =  1,000  x  1.198  cos  co ; 

hence, 

cos  w  =  .2365, 

o>  =  76.3°, 

and  the  angle  of  hfsteretic  advance  of  phase, 
a  =  90°  -  u  =  13.7°. 


EQUIVALENT  SINE    WAVES.  119 

The  magnetic  energy  current  is  then, 

*'  cos  co  =  .283  ; 
the  magnetizing  current  ; 

/'sin  w  =  1.165. 
Adding  the  instantanous  values  of  E.M.F.  e0  in  column 

(4),    gives    14,648,    thus    the    mean    value,  HL648  =  813.8. 

18 
Since  the  effective  value  is_  1,000,  the  mean  value  of  a  sine 

9  V^ 

wave  would  be,  1,000-    -  =  904,  hence  the  form  factor  is, 


9 

7T 

904 


813.8        '- 

Adding  the  instantaneous  values  of  current  i  in  column 
(10),  irrespective  of  their  sign,  gives  17.17,  thus  the  mean 

17  17 
value,  _J__=.954.     Since  the  effective  value  =  1.198,  the 

form  factor  is, 

1.198  ^/2_106 
.954      TT 

The  instantaneous  values  of  E.M.F.  eot  current  i  induc- 
tion (B  and  power  /  are  plotted  in  Fig.  37,  their  correspond- 
ing sine  waves  in  Fig.  38. 


PART    II. 

SPECIAL    APPARATUS. 


INTRODUCTION. 

BY  the  direction  of  the  energy  transmitted,  electric 
machines  have  been  divided  into  generators  and  motors.  By 
the  character  of  the  electric  power  they  have  been  dis- 
tinguished as  direct  current  and  as  alternating  current 
apparatus. 

With  the  advance  of  electrical  engineering,  however, 
these  sub-divisions  have  become  unsatisfactory  and  insuf- 
ficient. 

The  division  into  generators  and  motors  is  not  based  on 
any  characteristic  feature  of  the  apparatus,  and  is  thus  not 
rational.  Practically  any  electric  generator  can  be  used  as 
motor,  and  conversely,  and  frequently  one  and  the  same  ma- 
chine is  used  for  either  purpose.  Where  a  difference  is  made 
in  the  construction,  it  is  either  only  quantitative,  as,  for 
instance,  in  synchronous  motors  a  much  higher  armature  re- 
action is  used  than  in  synchronous  generators,  or  it  is  in 
minor  features,  as  direct  current  motors  usually  have  only 
one  field  winding,  either  shunt  or  series,  while  in  generators 
frequently  a  compound  field  is  employed.  Furthermore, 
apparatus  have  been  introduced  which  are  neither  motors 
nor  generators,  as  the  synchronous  machine  producing  watt- 
less lagging  or  leading  current,  etc.,  and  the  different  types 
of  converters. 

The  sub-division  into  direct  current  and  alternating  cur- 

121 


122  ELECTRICAL   ENGINEERING. 

rent  apparatus  is  unsatisfactory,  since  it  includes  in  the  same 
class  apparatus  of  entirely  different  character,  as  the  induc- 
tion motor  and  the  alternating  current  generator,  or  the 
constant  potential  commutating  machine,  and  the  rectifying 
arc  light  machine. 

Thus  the  following  classification,  based  on  the  charac- 
teristic features  of  the  apparatus,  has  been  adopted  by  the 
A.  I.  E.  E.  Standardizing  Committee,  and  is  used  in  the  fol- 
lowing discussion.  It  refers  only  to  the  apparatus  trans- 
forming between  electric  and  electric,  and  between  electric 
and  mechanical  power. 

1st.  Commutating  machines,  consisting  of  a  uni-direc- 
tional  magnetic  field  and  a  closed  coil  armature,  connected 
with  a  multi-segmental  commutator. 

2d.  Synchronous  machines,  consisting  of  a  uni-direc- 
tional  magnetic  field  and  an  armature  revolving  relatively  to 
the  magnetic  field  at  a  velocity  synchronous  with  the  fre- 
quency of  the  alternating  current  circuit  connected  thereto. 

3d.  Rectifying  apparatus ;  that  is,  apparatus  reversing 
the  direction  of  an  alternating  current  synchronously  with 
the  frequency. 

4th.  Induction  machines,  consisting  of  an  alternating 
magnetic  circuit  or  circuits  interlinked  with  two  electric  cir- 
cuits or  sets  of  circuits,  moving  with  regard  to  each  other. 

5th.  Stationary  Induction  apparatus,  consisting  of  a 
magnetic  circuit  interlinked  with  one  or  more  electric 
circuits. 

6th.      Electrostatic    and    electrolytic    apparatus,    as    con* 
densers  and  polarization  cells. 

Apparatus  changing  from  one  to  a  different  form  of 
electric  energy  have  been  defined  as  : 

A.  —  Transformers,  when  using  magnetism,  and  as 

B.  —  Converters,  when  using  mechanical  momentum  as 
intermediary  form  of  energy. 

The  transformers  as  a  rule  are  stationary,  the  converters 
rotary  apparatus.  Motor-generators .  transforming  from 


INTRODUCTION.  123 

electrical  over  mechanical  to  electric  power  by  two  separate 
machines,  and  dynamotors,  in  which  these  two  machines  are 
combined  in  the  same  structure,  are  not  included  under  con- 
verters. 

1 .  Commutating  machines  as  generators  are  usually  built 
to  produce  constant  potential  for  railway,  incandescent  light- 
ing, and  general  distribution.      Only  rarely  they  are  designed 
for  approximately  constant   power  for  electro- metallurgical 
work,  or  approximately  constant    current  for  series  incan- 
descent or  arc  lighting.      As  motors  commutating  machines 
give  approximately   constant    speed,  —  shunt  motors,  —  or 
large  starting  torque,  —  series  motors. 

When  inserted  in  series  in  a  circuit,  and  controlled  so  as 
to  give  an  E.M.F.  varying  with  the  conditions  of  load  on  the 
system  these  machines  are  "  boosters"  and  are  generators 
when  raising  the  voltage,  and  motors  when  lowering  it. 

Commutating  machines  may  be  used  as  direct  current 
converters  by  transforming  power  from  one  side  to  the  other 
side  of  a  three-wire  system. 

2.  While  in  commutating  machines  the  magnetic  field 
is   almost    always    stationary    and   the    armature    rotating, 
synchronous  machines  are  built  with  stationary  field  and  re- 
volving armature,  or  with  stationary  armature  and  revolving 
field,  or  inductor  machines   with   stationary  armature   and 
stationary  field  winding,  but  revolving  magnetic  circuit. 

By  the  number  and  character  of  the  alternating  circuits 
connected  to  them  they  are  single-phase  or  polyphase  ma- 
chines. As  generators  they  comprise  practically  all  single- 
phase  and  polyphase  alternating  current  generators,  as 
motors  a  very  important  class  of  apparatus,  the  synchronous 
motors,  which  are  usually  preferred  for  large  powers, 
especially  where  frequent  starting  and  considerable  starting 
torque  are  not  needed.  Synchronous  machines  may  be  used 
as  compensators  to  produce  wattless  current,  leading  by  over- 
excitation,  lagging  by  under-excitation,  or  may  be  used  as 
phase  converters  by  operating  a  polyphase  synchronous 


124  ELECTRICAL   ENGINEERING. 

motor  by  one  pair  of  terminals  from  a  single-phase  circuit. 
The  most  important  class  of  converters,  however,  are  the 
synchronous  commutating  machines,  to  which,  therefore,  a 
special  chapter  will  be  devoted  in  the  following. 

Synchronous  commutating  machines  contain  a  imi-direc- 
tional  magnetic  field  and  a  closed  circuit  armature  connected 
simultaneously  to  a  segmental  direct  current  commutator 
and  by  collector  rings  to  an  alternating  circuit,  mostly  a 
polyphase  system.  These  machines  thus  can  either  receive 
alternating  and  yield  direct  current  power  as  synchronous 
converters,  or  simply  "converters,"  or  receive  direct  and  yield 
alternating  current  power  as  inverted  converters,  or  driven 
by  mechanical  power  yield  alternating  and  direct  current  as 
double  current  generators.  Or  they  can  combine  motor  and 
generator  action  with  their  converter  action.  Thus  a  com- 
mon combination  is  a  synchronous  converter  supplying  a 
certain  amount  of  mechanical  power  as  synchronous  motor. 

3.  Rectifying  Machines  are  apparatus  which  by  a  syn- 
chronously revolving  rectifying  commutator  send  the  succes- 
sive half  waves  of  an  alternating  single-phase  or  polyphase 
circuit  in  the  same  direction  into  the  receiving  circuit.     The 
most  important  class  of  such  apparatus  are  the  open  coil 
arc  light  machines,  which  generate  the  rectified  E.M.F.  at 
approximately  constant  current,  in  a  star-connected  three- 
phase  armature  in  the  Thomson-Houston,  as  quarter-phase 
E.M.F.  in  the  Brush  arc-light  machine. 

4.  Induction    machines   are   generally   used  as  motors, 
polyphase  or  single-phase.     In  this  case  they  run  at  practi- 
cally constant  speed,  slowing  down  slightly  with  increasing 
load.     As  generators  the  frequency  of  the  E.M.F.  supplied 
by  them  differs  from    and  is  lower  than  the  frequency  of 
rotation,  but  their  operation  depends  upon  the  phase  rela- 
tion of  the  external  circuit.     As  phase  converters  induction 
machines  can  be  used  in  the  same  manner  as  synchronous 
machines.     Their  most  important  use  besides  as  motors  is, 
however,  as  frequency  converters,  by  changing  from  an  im- 


INTR  OD  UC  TION.  125 

pressed  primary  polyphase  system  to  a  secondary  polyphase 
system  or  different  frequency.  In  this  case  when  lowering 
the  frequency,  mechanical  energy  is  also  produced ;  when 
raising  the  frequency,  mechanical  energy  is  consumed. 

5.  The  most  important  stationary  induction  apparatus  is 
the  transformer,  consisting  of  two  electric  circuits  interlinked 
with  the  same  magnetic  circuit.     When  using  the  same  or 
part  of  the  same  electric  circuit  for  primary  and  secondary, 
the  transformer  is  called  a  compensator  or  auto-transformer. 
When    inserted  in    series    into   an  alternating    circuit,  and 
arranged   to  vary   the  E.M.F.,    the   transformer   is    called 
potential  regulator  or  booster.     The  variation  of  secondary 
E.M.F.  may  be  secured  by  varying  the  relative  number  of 
primary    and  secondary  turns,    or   by  varying  the   mutual 
induction    between    primary   and  secondary    circuit,  either 
electrically  or  magnetically.     The  stationary  induction  appa- 
ratus with  one  electric  circuit  are  used  for  producing  wattless 
lagging  currents,  as  reactive  or  choking  coils. 

6.  Condensers  and  polarization   cells   produce  wattless 
leading    currents,    the  latter,    however,  at  a  very  low  effi- 
ciency, while  the  efficiency  of  the  condenser  is  extremely 
high,  frequently  above  99%  ;  that  is,  the  loss  of  energy  is  less 
than  1%  of  the  apparent  volt-ampere  input. 

To  this  classification  may  be  added  the  uni-polar>  or, 
more  correctly,  non-polar  machine,  in  which  a  conductor 
cuts  a  magnetic  field  at  a  uniform  rate.  Thus  far  these 
machines  do  not  appear  of  any  practical  value. 

Regarding  apparatus  transforming  between  electric  energy 
and  forms  of  energy  differing  from  electric  or  mechanical 
energy :  The  transformation  between  electrical  and  chemical 
energy  is  represented  by  the  primary  and  secondary  battery 
and  the  electrolytic  cell.  The  transformation  between  elec- 
trical and  heat  energy  by  the  thermopile,  and  the  electric 
heater  or  electric  furnace.  The  transformation  between 
electrical  and  light  energy  by  the  incandescent  and  arc 
lamps. 


126  ELECTRICAL   ENGINEERING. 


A.    SYNCHRONOUS    MACHINES. 
I.     General. 

THE  most  important  class  of  alternating  current  appa- 
ratus consists  of  the  synchronous  machines.  They  com- 
prise the  alternating  current  generators,  single-phase  and 
polyphase,  the  synchronous  motors,  the  phase  compensators 
and  the  exciters  of  induction  generators,  that  is,  synchronous 
machines  producing  wattless  lagging  or  leading  currents, 
and  the  converters.  Since  the  latter  combine  features  of 
the  commutating  machines  with  those  cf  the  synchronous 
machines  they  will  be  considered  separately. 

In  the  synchronous  machines  the  terminal  voltage  and 
the  induced  E.M.F.  are  in  synchronism  with,  that  is,  of  the 
same  frequency  as,  the  speed  of  rotation. 

These  machines  consist  of  an  armature,  in  which  E.M.F. 
is  induced  by  the  rotation  relatively  to  a  magnetic  field,  and 
a  continuous  magnetic  field,  excited  either  by  direct  current, 
or  by  the  reaction  of  displaced  phase  armature  currents,  or 
by  permanent  magnetism.  , 

The  formula  of  induction  of  synchronous  machines,  or 
commonly  called  alternators,  is, 

E  =  V2  WV>*$  =  4.44  Mi®, 

where  n  is  the  number  of  armature  turns  in  series  interlinked 
with  the  magnetic  flux  3>  (in  ml.  per  pole),  and  N  the  fre- 
quency of  rotation  (in  hundreds  cf  cycles  per  sec.),  E  the 
E.M.F.  induced  in  the  armature  turns. 

This  formula  assumes  a  sine  wave  of  E.M.F.  If  the 
E.M.F.  wave  differs  from  sine  shape,  the  E.M.F.  is 


SYNCHRONOUS  MACHINES.  127 

2  V2 
where  y  —  form  factor  of  the  wave,  or  -  -  times  ratio  of 

7T 

effective  to  mean  value  of  wave. 

The  form  factor  y  depends  upon  the  wave  shape  of  the 
induced  E.M.F.  The  wave  shape  of  E.M.F.  induced  in 
a  single  conductor  on  the  armature  surface  is  identical  with 
that  of  the  distribution  of  magnetic  flux  at  the  armature 
surface  and  will  be  discussed  more  fully  in  the  chapter  en 
Commutating  Machines.  The  wave  of  total  E.M.F.  is  the 
sum  of  the  waves  of  E.M.F.  in  the  individual  conductors, 
added  in  their  proper  phase  relation,  as  corresponding  to 
their  relative  positions  on  the  armature  surface. 

In  a  Y  or  star-connected  three-phase  machine,  if  EQ  = 
E.M.F.  per  circuit,  cr  Y  cr  star  E.M.F.,  E  =  -E0^/&  is 
the  E.M.F.  between  terminals  or  A  cr  ring  E.M.F.,  since 
two  E.M.F.'s  displaced  by  60°  are  connected  in  series  be- 
tween terminals  (V3  =  2  cos  60°). 

In  a  A-connected  three-phase  machine,  the  E.M.F.  per 
circuit  is  the  E.M.F.  between  the  terminals,  or  A  E.M.F. 

In  a  F-connected  three-phase  machine,  the  current  per 
circuit  is  the  current  issuing  from  each  terminal,  cr  the  line 
current,  or  Y  current. 

In  a  A-connected  three-phase  machine,  if  70  =  current  per 
circuit,  or  A  current,  the  current  issuing  from  each  terminal, 
or  the  line  or  Y  current,  is 


Thus  in  a  three-phase  system,  A  current  and  E.M.F., 
and  'Y  current  and  E.M.F.  (or  ring  and  star  current  and 
E.M.F.  respectively),  are  to  be  distinguished.  They  stand 
in  the  proportion  :  1  -s-  V3. 

As  a  rule,  when  speaking  of  current  and  of  E.M.F  in  a 
three-phase  system,  under  current  the  Y  current  or  current 
per  line,  and  under  E.M.F.,  the  ^A  E.M.F.  or  E.M.F. 
between  lines  is  understood. 


128  ELECTRICAL   ENGINEERING. 

II.     E.M.F.'s. 

In  a  synchronous  machine  we  have  to  distinguish  be- 
tween terminal  voltage  £,  real  induced  E.M.F.  Ev  virtual 
induced  E.M.F.  Ey  and  nominal  induced  E.M.F.  E0. 

The  real  induced  E.M.F.  El  is  the  E.M.F.  induced  in 
the  alternator  armature  turns  by  the  resultant  magnetic 
flux,  or  magnetic  flux  interlinked  with  them,  that  is,  by  the 
magnetic  flux  passing  through  the  armature  core.  It  is 
equal  to  the  terminal  voltage  plus  the  E.M.F.  consumed  by 
the  resistance  of  the  armature,  these  two  E.M.F.'s  being 
taken  in  their  proper  phase  relation, 


where  /  =  current  in  armature,  r  =  effective  resistance. 

The  virtual  induced  E.M.F.  E2  is  the  E.M.F.  which 
would  be  induced  by  the  flux  produced  by  the  field  poles,  or 
flux  corresponding  to  the  resultant  M.M.F.,  that  is,  the  resul- 
tant of  the  M.M.F.'s  of  field  excitation  and  of  armature 
reaction.  Since  the  magnetic  flux  produced  by  the  arma- 
ture, or  flux  of  armature  self-induction,  combines  with  the 
field  flux  to  the  resultant  flux,  the  flux  produced  by  the  field 
poles  does  not  pass  through  the  armature  completely,  and 
the  virtual  E.M.F.  and  the  real  induced  E.M.F.  differ  from 
each  other  by  the  E.M.F.  of  armature  self-induction;  but 
the  virtual  induced  E.M.F.,  as  well  as  the  E.M.F.  induced  in 
the  armature  by  self-induction,  have  no  real  and  independent 
existence,  but  are  merely  fictitious  components  of  the  real 
or  resultant  induced  E.M.F.  Er 

The  virtual  induced  E.M.F.  is, 


where  x  is  the  self-inductive  armature  reactance,  and  the 
E.M.F.  consumed  by  self-induction  /r,  is  to  be  combined 
with  the  real  induced  E.M.F.  El  in  the  proper  phase  relation.. 
The  nominal  induced  E.M.F.  E0  is  the  E.M.F.  which 
would  be  induced  by  the  field  excitation  if  there  were  neither 


SYNCHRONOUS  MACHINES.  129 

self-induction  nor  armature  reaction,  and  the  saturation  were 
the  same  as  corresponds  to  the  real  induced  E.M.F.  It 
thus  does  not  correspond  to  any  magnetic  flux,  and  has  no 
existence  at  all,  but  is  merely  a  fictitious  quantity,  which, 
however,  is  very  useful  for  the  investigation  of  alternators 
by  allowing  the  combination  of  armature  reaction  and  self- 
induction  into  a  single  effect,  by  a  (fictitious)  self-induction 
or  synchronous  reactance  ;r0.  The  nominal  induced  E.M.F. 
would  be  the  terminal  voltage  at  open  circuit  and  full  load 
excitation  if  the  saturation  curve  were  a  straight  line. 

The  synchronous  reactance  XQ  is  thus  a  quantity  com- 
bining armature  reaction  and  self-induction  of  the  alternator. 
It  is  the  only  quantity  which  can  easily  be  determined  by 
experiment,  by  running  the  alternator  on  short  circuit  with 
excited  field.  If  in  this  case  70  =  current,  WQ  =  loss  of  power 
in  the  armature  coils,  EQ  =  E.M.F.  corresponding  to  the  field 

J7 

excitation  at  open  circuit  ;    -y  =  ZQ  is  the  synchronous  im- 

^o 
W 
pedance  -j~2-  =r0  is  the  effective  resistance  (ohmic  resistance 


plus  load  losses),  and  ;r0  =  V^r02  —  r02  the  synchronous  re- 
actance. 

In  this  feature  lies  the  importance  of  the  term  4<  nominal 
induced  E.M.F."  EQ: 

these  terms  being  combined  in  their  proper  phase  relation. 
In  a  polyphase  machine,  these  considerations  apply  to  each 
of  the  machine  circuits  individually. 

III.     Armature  Reaction. 

The  magnetic  flux  in  the  field  of  an  alternator  under 
load  is  produced  by  the  resultant  M.M.F.  of  the  field  excit- 
ing current  and  of  the  armature  current.  It  depends  upon 
the  phase  relation  of  the  armature  current.  The  E.M.F. 
induced  by  the  field  exciting  current  or  the  nominal  in 


130 


ELECTRICAL   ENGINEERING. 


duced  E.M.F.  reaches  a  maximum  when  the  armature  coil 
faces  the  position  midway  between  the  field  poles,  as  shown 
in  Fig.  39,  A  and  A'.  Thus,  if  the  armature  current  is  in 


GENERATOR 


Fig.  39. 


phase  with  the  nominal  induced  E.M.F.,  it  reaches  its  maxi- 
mum, in  the  same  position  A,  A'  of  armature  coil  as  the 
nominal  induced  E.M.F.,  and  thus  magnetizes  the  preceding, 
demagnetizes  the  following  magnet  pole  (in  the  direction  of 
rotation)  in  an.  alternating  current  generator  A  ;  magnetizes 
the  following  and  demagnetizes  the  preceding  magnet  pole 
in  a  synchronous  motor,  A'  (since  in  a  generator  the  rotation 
is  against,  in  a  synchronous  motor  with  the  magnetic 
attractions  and  repulsions  between  field  and  armature).  In 
this  case  the  armature  current  neither  magnetizes  nor 
demagnetizes  the  field  as  a  whole,  but  magnetizes  the  one 
side,  demagnetizes  the  other  side  of  each  field  pole,  and  thus 
merely  distorts  the  magnetic  fielA 

If  the  armature  current  lags  behind  the  nominal  induced 
E.M.F.,  it  reaches  its  maximum  in  a  position  where  the 
armature  coil  already  faces  the  next  magnetic  pole,  as  shown 
in  Fig.  39,  B  and  B',  and  thus  demagnetizes  the  field  in  a 
generator,  B,  magnetizes  it  in  a  synchronous  motor,  B'. 


6"  YNCHR  ONO  US  MA  CHINES. 


131 


If  the  armature  current  leads  the  nominal  induced 
E.M.F.,  it  reaches  its  maximum  in  an  earlier  position,  while 
the  armature  coil  still  partly  faces  the  preceding  magnet 
pole,  as  shown  in  Fig.  39,  C  and  C't  and  thus  magnetizes  the 
field  in  a  generator,  Fig.  39,  C,  and  demagnetizes  it  in  a 
synchronous  motor,  C'. 

With  non-inductive  load,  or  with  the  current  in  phase 
with  the  terminal  voltage  of  an  alternating  current  generator, 
the  current  lags  behind  the  nominal  induced  E.M.F.,  due  to 
armature  reaction  and  self-induction,  and  thus  partly 
demagnetizes ;  that  is,  the  voltage  is  lower  under  load  than 
at  no  load  with  the  same  field  excitation.  That  is,  lagging 
current  demagnetizes,  leading  current  magnetizes,  the  field 
of  an  alternating  current  generator,  while  the  opposite  is  the 
case  with  a  synchronous  motor. 

In  Fig.  40  let  ~OF  =  $  =  resultant  M.M.F.  of  field  excita- 
tion and  armature  current  (the  M.M.F.  of  the  field  excitation 
being  alternating  also  with 
regard  to  the  armature 
coil,  due  to  its  rotation) 
and  w2  the  lag  of  the  cur- 
rent /  behind  the  virtual 
E.M.F.  E2  induced  by  the 
resultant  M.M.F. 

The  virtual  E.M.F.  E2 
lags  90°  behind_the  re- 
sultant flux  of  OF,,2ind  is 
thus  represented  by  OE2 
in  Fig.  40,andthe'M.M.F. 
of  the  armature  current 
/  by  Of,  lagging  by  angle 
o>2  behind  OFV  The  re- 
sultant M.M.F.  OS  is  the 
diagonal  of  the  parallelo- 
gram with  the  component  M.M.F  's  Of  —  armature  M.M.F. 
and  O$Q  =  total  impressed  E.M.F.  or  field  excitation,  as 


132  ELECTRICAL   ENGINEERING. 

sides,  and  from  this  construction  O$Q  is  found.  O$Q  is  thus 
the  position  of  the  field  pole  with  regard  to  the  armature. 
It  is  trigonometrically, 


<F0  =  ViF2  +/2  +  2  &/  sin  (?2. 

If  /=  current  per  armature  turn  in  amperes  effective, 
n  =  number  of  turns  per  pole  in  a  single-phase  alternator, 
the  armature  reaction  is/=  TZ/  ampere  turns  effective,  and 
is  pulsating  between  zero  and  ;/  /  V2. 

In  a  quarter-phase  alternator  with  n  turns  per  pole,  and 
phase  in  series  and  /  amperes  effective  per  turn,  the  arma- 
ture reaction  per  phase  is  n  I  ampere  turns  effective,  and 
n  I  V2  ampere  turns  maximum.  The  two  phases  magnetize 
in  quadrature,  in  phase,  and  in  space.  Thus  at  the  time  /, 
corresponding  to  angle  <f>  after  the  maximum  of  the  first 
phase,  the  M.M.F.  in  the  direction  by  angle  <f>  behind  the 
direction  of  the  magnetization  of  the  first  phase  is  nl  V2  cos2 
<f>.  The  M.M.F.  of  the  second  phase  is  n  I  V2  sin2  <f>,  thus 
the  total  M.M.F.  or  the  armature  reaction/  =  w/V2,  and  is 
constant  in  intensity,  but  revolves  synchronously  with  regard 
to  the  armature  ;  that  is,  it  is  stationary  with  regard  to  the 
field. 

In  a  three-phaser  of  n  turns  in  series  per  pole,  and  phase 
and  /amperes  effective  per  turn,  the  M.M.F.  of  each  phase 
is  «/V2  ampere  turns  maximum,  thus  at  angle  <f>  in  position 
and  angle  <f>  in  time  behind  the  maximum  of  one  phase  ; 

the  M.M.F.  of  this  phase  is, 


The  M.M.F.  of  the  second  phase  is, 
«/V2cos2O-f-  120)  =  «/V2(-.5cos<£-.5  V3sin<£)2. 
The  M.M.F.  of  the  third  phase  is, 
«/V2cos2(<£  -h  240)  =  /*/V2(-.5cos<£  +.5  V3sin</>)2. 
Thus  the  total  M.M.F.  or  armature  reaction, 
/=  >z/V2(cos2<£  -f.25cos2<£ 
=  1.5*7  V2, 


SYNCHRONOUS  MACHINES.  133 

constant  in  intensity,  but  revolving  synchronously  with 
regard  to  the  armature,  that  is,  stationary  with  regard  to  the 
field.  These  values  of  armature  reaction  correspond  strictly 
only  to  the  case  where  all  conductors  of  the  same  phase  are 
massed  together  in  one  slot.  If  the  conductors  of  each  phase 
are  distributed  over  a  greater  part  of  the  armature  surface, 
the  values  of  armature  reaction  have  to  be  multiplied  by  the 
average  cosine  of  the  total  angle  of  spread  of  each  phase. 

IV.     Self-induction. 

The  effect  of  self-induction  is  similar  to  that  of  armature 
reaction,  and  depends  upon  the  phase  relation  in  the  same 
manner. 

If  E^=  real  induced  voltage,  0^=  lag  of  current  behind 
induced  voltage  Ev  the  magnetic  flux  produced  by  the  arma- 
ture  current  /  is  in  phase  with  the  current,  and  thus  the 
counter  E.M.F.  of  self-induction  in  quadrature  behind  the 
current,  and  therefore  the  E.M.F.  consumed  by  self-induction 
in  quadrature   ahead  of  the  current.     Thus  in  Fig.  41,  de- 
noting   EO1  =  El   the    induced 
E.M.F.,  the  current  is,  Of  =  /, 
lagging  Wj behind  OEV  the  E.M.F. 
consumed    by    self-induction 
OE",  90°  ahead  of  the  current, 
and  the  virtual  induced  E.M.F. 
Ev  the  resultant  OS1  and  OE{' . 
As  seen,  the  diagram  of  E.M.F.'s 
of    self-induction    is    similar   to 
the  diagram  of  M.M.F.'s  of  armature  reaction. 

From  this  diagram  we  get  the  effect  of  load  and  phase 
relation  upon  the  E.M.F.  of  an  alternating  current  generator. 

Let  E  =  terminal  voltage  per  machine  circuit, 

/  =  current  per  machine  circuit, 
and         <o  =  lag  of  current  behind  terminal  voltage. 
Let   r  =  resistance, 

x  =  reactance  of  alternator  armature. 


134 


ELECTRICAL   ENGINEERING. 


Then,  in  the  polar  diagram,  Fig.  42, 

OE  =  E,  the  terminal  voltage,  assumed  as  zero  vector. 

Of  =  f,    the  current,  lagging  by  angle  EOf =  co. 

The  E.M.F.  con- 
sumed by  resistance 
is, 

OJ&i  =  Ir  in  phase 
with  Of. 

The  E.M.F.  con- 
sumed by  react- 
ance, 

O£2f=  fx,    90° 
ahead  of  Of. 

The  real  induced   E.M.F.  is  found  by 
combining  OE  and  OE{  to, 


The    virtual    induced    E.M.F.    is 
and  OE%  combined  to 


The  M.M.F.  required  to  produce  this  E.M.F.  Ey  is  OF 
=  &,  90°  ahead  of  OEV     It   is  the  resultant  of  armature 


o il. 


E      Ei 


Fig.  43. 


M.M.F.  or  armature  reaction,  and  of  impressed  M.M.F.  or 
field  excitation.  The  armature  M.M.F.  is  in  phase  with  the 
current  /,  and  is  nl  in  a  single-phaser,  n  V2  /  in  a  quarter- 


SYNCHRONOUS  MACHINES. 


135 


phaser,  1.5  V2  ;//in  a  three-phaser,  if  n  —  number  of  arma- 
ture turns  per  pole  and  phase.  The  M.M.F.  of  armature 
reaction  is  represented  in  the  diagram  by  Of  —  fm  phase 
with  Of,  and  the  impressed  M.M.F.  or  field  excitation  G$Q 
=  SQ  is  the  side  of  a  parallelogram  with  6^  as  diagonal  and 
Of  as  other  side.  Or  the  M.M.F.  consumed  by  armature 
reaction  is  represented  by  Ofr  =  f  in  opposition  to  OI. 
Combining  Of  and  OS  gives  G$Q=  Sv  the  field  excitation. 

0 


Fig.  45. 


In  Figs.  43,  44,  45  are  drawn  the  diagrams 
for  w  =  zero  or  non-inductive  load,  to  =  60°,  or 
60°  lag  (inductive  load  of  power  factor  .50),  and 
a,  =  -  60°,  or  60°  lead  (anti-inductive  load  of 
power  factor  .50). 

As  seen,  with  the  same  terminal  voltage  E9 
with  inductive  load  a  much  higher,  with  anti- 
inductive  load  a  lower,  field  excitation  SF0  is  re- 
quired than  with  non-inductive  load.  At  open 
circuit  the  field  excitation  required  to  produce 

terminal  voltage  E  would  be-^^  =  $w  or  less  than  the  field 

excitation  £F0  with  non-inductive  load. 

Inversely,  at  constant  field  excitation,  the  voltage  of  an 
alternator  drops  at  non-inductive  load,  drops  much  more  with 
inductive  load,  and  drops  less,  or  even  rises,  with  anti- 
inductive  load. 


136 


ELECTRICAL   ENGINEERING. 


V.     Synchronous  Reactance. 

In  general,  both  effects,  armature  self-induction  and 
armature  reaction,  can  be  combined  by  the  term  "  synchro- 
nous reactance." 

Let  r  =  effective  resistance, 

XQ  =  synchronous  reactance  of  armature,  as  discussed  in 

section  II. 
Let  E  =  terminal  voltage, 

/=  current, 

(o  =  angle  of  lag  of  current  behind  terminal  voltage. 


Fig.  46. 


Fig.  47. 


It  is  in  polar  diagram,  Fig.  46. 

OE  =  E  =  terminal  voltage  assumed  as  zero  vector. 
OI  =  I  —  current  lagging  by  angle  EOI  =  <o  behind  the  ter- 
minal voltage. 

OEi  =  Ir  is  the  E.M.F.  consumed  by  resistance,  in 
phase  with  OI,  and  OE^  =  Ix^  the  E.M.F.  consumed  by  the 
synchronous  reactance,  90°  ahead  of  the  current  OI. 

OE{  and  OE{  combined  give  OE'  =  E'  the  E.M.F. 
consumed  by  the  synchronous  impedance. 

Combining  OE{,  OE0f,  OE  gives  the  nominal  induced 
E.M.F.  OEQ=  Ew  corresponding  to  the  field  excitation  9^. 

In  Figs.  47,  48,  49,  are  shown  the  diagrams  for  <o  =  O  or 
non-inductive  load,  o>  =  60°  lag  or  inductive  load,  and  w  = 
—  60°  or  anti-inductive  load. 

Resolving  all  E.M.F.'s  in  components  in  phase  and  in 


SYNCHRONOUS  MACHINES. 


137 


quadrature  with  the  current,  or  in  energy  components  and 
in  wattless  components,  it  is,  in  symbolic  expression, 

Terminal  voltage  E  =  E  cos  o>  —jE  sin  w, 

E.M.F.  consumed  by  resistance,  -£/  =  tr, 

E.M.F.  consumed  by  synchronous  reactance,  EQ  =  —  jix§, 

Nominal  induced  E.M.F., 

EQ  =  E  4-  EI  -\-  EQ '=  (JScos  <o  4-  ir)—j(E  sin  w 

or,  since 

/      energy  current\ 

cos  w  =  />  =  power  factor  of  load    = =r — 

\       total  current  / 

and 

^  =  Vl  — /2=  sin  w  =  inductance  factor  of  load 
wattless  current\ 


it  is, 


total  current    / 

EQ  =  (Ep  4-  /r)  -y  (^  4-  ^0), 
or,  absolute, 


hence, 


4-  ir)a  4-  (Eq  4- 


f /ST.  48. 


The  power  delivered  by  the  alternator  into  the  external 
circuit  is, 

P  =  iEp, 
that  is,  current  times  energy  component  of  terminal  voltage. 

The  electric    power  produced  in    the  alternator    arma- 
ture is, 


138 


ELECTRICAL  ENGINEERING 


that  is,  current  times  energy  component  of  nominal  induced 
E.M.F.,  or,  what  is  the  same,  current  times  energy  component 
of  real  induced  E.M.F. 


SYNCHRONOUS   GENERATOR 


COMPOUNDING  CURVES 
E=1OOO 


r-.i     SS-ff. 


2,0      40      60      80      100     120     1-fO     160     180     2QO 

Fig.  50. 


YNCHR  OArO  US  MA  CHINES. 


139 


VI.    Characteristic  Curves  of  Alternating  Current  Generator. 

In  Fig.  50  are  shown,  at  constant  terminal  voltage  E,  the 
values  of  nominal  induced  E.M.F.  £0,  and  thus  of  field  exci- 
tation SF,,,  with  the  current  /  as  abscissae  and  for  the  three 
conditions, 

1).    Non-inductive  load,/  —  1,  q  =  0. 

2).    Inductive  load  of  o>  =  60°  lag,/  =  .5,  q  =  .866. 

3).    Anti-inductive  load  of— w^=600  lead,/ =  .5,  ^=—.866. 


SYNCHRONOUS  GENERATOR. 


ON  CURN 


E 
150O 


REGULATI 


•.1 


12OO 


10OO 


3OO 


600 


200 


20   4,0.  60   80   10O 


1  20/1 ' 


0  16O  180  2OO  220  240  260  28O\ 


Fig.  51. 

The  values  r  =  .1,  XQ=  5,  E  =  1000,  are  assumed.  These 
curves  are  called  the  "  Compounding  Curves  of  the  Synchro- 
nous Generator." 

In  Fig.  51  are  shown,  at  constant  nominal  induced 
E.M.F.  EQ,  that  is,  constant  field  excitation  &v  the  values  of 


140 


ELE C TR1CAL   ENGINEERING. 


terminal  voltage  E  with  the  current  /  as  abscissae  and  for 
the  same  resistance  and  synchronous  reactance  r  —  .1,  ,t'0=  5, 
for  the  three  different  conditions  : 

1).    Non-inductive  load,/  =  1,   q  =  0,   £()=  1,127. 
2).    Inductive  load  of  60°  lag, 

/  =  .5,   ^  =  .866,   £0  =  1,458. 
3).    Anti-inductive  load  of  60°  lead, 

/=.5,    ?  =  -.866,  £0  =  628. 

The  values  of  EQ  (and  thus  of  £F0)  are  assumed  so  as  to 
give  £  =  1000  at  /=  100.  These  curves  are  called  the 
"Field  Characteristics"  of  the  alternator,  or  the  "Regula- 
tion Curves  "  of  the  synchronous  generator. 


20   40   60   80   100  120  140  16O  18O  2CO  220  240  260  ,280 
Fig.  52. 


SYNCHRONOUS  MACHINES.  141 

In  Fig.  52  are  shown  the  "Load  Curves"  of  the  ma- 
chine, with  the  current  /  as  abscissae  and  the  watts  output 
as  ordinates  corresponding  to  the  same  three  conditions  as 
Fig.  51.  From  the  field  characteristics  of  the  alternator 
are  derived  the  open-circuit  voltage  of  1127  at  full  non- 
inductive  load  excitation,  =  1.127  times  full  load  voltage; 
the  short-circuit  current  225  at  full  non-inductive  load  exci- 
tation, =  2.25  times  full-load  current;  and  the  maximum  out- 
put, 124  K.W.,  at  full  non-inductive  load  excitation,  =  1.24 
times  rated  output,  at  voltage  775  and  current  160  amperes. 
On  the  point  of  the  field  characteristic  on  which  the  alter- 
nator works  depends  whether  it  tends  to  regulate  for,  that 
is,  maintain,  constant  voltage,  or  constant  current,  or  con- 
stant power. 

VII.    Synchronous  Motor. 

As  seen  in  the  preceding,  in  an  alternating  current 
generator  the  field  excitation  required  for  a  given  terminal 
voltage  and  current  depends  upon  the  phase  relation  of  the 
external  circuit  or  the  load.  Inversely  in  a  synchronous 
motor,  the  phase  relation  of  the  current 
flowing  into  the  armature  at  a  given 
terminal  voltage  depends  upon  the  field 
excitation  and  the  load. 

Thus,  if  E  =  terminal  voltage  or 
impressed  E.M.F.,  /  =  current,  <«>  =  lag 
of  current  behind  impressed  E.M.F.  in 
a  synchronous  motor  of  resistance  r 
and  synchronous  reactance  x^  the  polar 
diagram  is  as  follows,  Fig.  58. 

OE  =  E  is  the  terminal  voltage  assumed  as  zero  vector. 
The  current  Of  =  7  lags  by  angle  EOI  =  <o.  _ 

The  E.M.F.  consumed  by  resistance,  is  OE\=Ir.  The 
E.M.F.  consumed  by  synchronous  reactance,  OJ50'=  Ixv 
Thus,  combining  OE{  and  OZ?0' gives  OE',  the  E.M.F.  con- 
sumed by  the  synchronous  impedance.  The  E.M.F.  con- 

I 


142 


ELE C  TRICA  L   ENGINEERING. 


sumed  by  the  synchronous  impedance  OE',  and  the  E.M.F. 
consumed  by  the  nominal  induced  or  counter  E.M.F.  of  the 
synchronous  motor  OEV  combined,  give  the  impressed 
E.M.F.  OE.  Hence  OEQ  is  one  side  of  a  parallelogram, 
with  OE^  as  the  other  side,  and  OE  as  diagonal.  OEW  equal 
and  opposite  OEV  would  thus  be  the  nominal  counter 
E.M.F.  of  the  synchronous  motor. 

In   Figs.  54,   55,   56,  are   shown  the  polar  diagrams  of 
the  synchronous  motor  for  <*>  =  0°,  <o  =  60°,  o>  =  —  60°,    As 


E' 


Fig.  56. 

seen,  the  field  excitation  has  to  be  higher  with  leading, 
lower  with  lagging  current  in  a  synchronous  motor,  while 
the  opposite  is  the  case  in  an  alternating  current  generator. 
In  symbolic  representation,  by  resolving  all  E.M.F.'s  in 
energy  components  in  phase  with  the  current  and  wattless 
components  in  quadrature  with  the  current  i,  we  have, 


SYNCHRONOUS  MACHINES.  143 

Terminal  voltage,  E  —  E  cos  o>  —jE  sin  w  =  ^/>  —jEq. 

E.M.F.  consumed  by  resistance,  E^  =  ir. 

E.M.F.  consumed  by  synchronous  reactance,  EJ  =  —  jixQ. 


Thus  the  E.M.F.  consumed  by  nominal  induced  E.M.F., 
or  motor  counter  E.M.F. 

EQ  =  E  —  EI   —  EQ  =  (E  cos  GO  —  ir)  —  j  (E  sin  w  —  /*0) 

=  (Ep-  ir)  -j(Eq  -  txL), 
or  absolute, 


E    =       JB  cos  (o  —  />2  +    jfi"  sin  w  — 


0 


~  ir? 
hence, 


The  power  consumed  by  the  synchronous  motor  is 
P=  iEp, 

that  is,  current  times  energy  component  of  impressed 
E.M.F. 

The  mechanical  power  delivered  by  the  synchronous 
motor  armature  is, 

PQ  =  i(Ep  -  ?>•), 

that  is,  current  times  energy  component  of  nominal  induced 
E.M.F.  Obviously  to  get  the  available  mechanical  power, 
the  power  consumed  by  mechanical  friction  and  by  molecular 
magnetic  friction  or  hysteresis,  and  the  power  of  field  excita- 
tion, has  to  be  subtracted  from  this  value  P0. 

VIII.   Characteristic  Curves  of  Synchronous  Motor. 

In  F*ig.  57  are  shown,  at  constant  impressed  E.M.F.  E 
the  nominal  counter  E.M.F.  EQ  and  thus  the  field  excitation 
SF0  required 

1.)    At  no  phase  displacement,  o>  =  0,  or  for  the  condition 

of  minimum  input ; 

2.)    For  to  =  H-  60,  or-60°  lag:    p  =  .5,  q  =  +  .866,  and 
3.)    For  to  =  -  60,  or  60°  lead :  /  =  .5,  q  -.  -  .866, 


144 


ELECTRICAL  ENGINEERING. 


with  the  current  /  as  abscissae,  for  the  constants, 
r=.l,  *0  =  5;  E  =  1,000. 


SYNCHRONOUS  MOTOR 
COMPOUNDING  CURVE. 


E=1000 


1900. 


800 


700 


600 


500 


400 


300 


200 


11OO 


800 


700 


600 


500 


400 


300 


200 


100 


20   40   60   80  100  120  140  160  180  200 

Fig.  57. 


These  curves  are  called  the  "  Compounding  Curves  of 
the  Synchronous  Motor." 

In  Fig.  58  are  shown,  with  the  power  output  Pl  =  i 


S  YNCHK  ONO  US  MA  CHINES. 


145 


(Ep  —  ir)  —  (hysteresis  and  friction)  as  abscissae,  and  the  same 
constants  r  =  .1,  x^  =  5,  E  =  1000,  and  constant  field  excita- 
tion JF0;  that  is,  constant  nominal  induced  or  counter  E.M.F. 
£0=1109  (corresponding  to /=!,  q  =  Oat  7=100),  the  values 


140 


SYNCHRONOUS  MO 
LOAD   CHARACTERS 


130 


120 


110 


LEAD* 


80 


80 


70 


60 


50 


50 


30 


30 


20 


20 


IP   20   80   40   50   60   7p   SO   9p  1QO  W.O  12O  13O  140  *•  }*• 
F/g.  5ff. 


of  current  /  and  power  factor  /.  As  hysteresis  loss  is 
assumed  3000  watts,  as  friction  2000  watts.  Such  curves 
are  called  "  Load  Characteristics  of  the  Synchronous  Motor." 
In  Fig.  59  are  shown,  with  constant  power  output,  P0= 
i  (Ep  —  ir),  and  the  same  constants  :  r  =  .1,  ;r0=  5,  E  —  1000, 
and  with  the  nominal  induced  voltage  E&  that  is  field  excita- 
tion £F0  as  abscissae,  the  values  of  current  7,  for  the  four  con- 
ditions : 


146  ELECTRICAL   ENGINEERING. 

P0  =      5  K.W.,  or  Pl  =     0  ,    or  no  load, 

P0  =    50  K.W.,  or  Pl  =    45  K.W.,   or  half  load, 

/»0  =    95  K.W.,  or  Pl  =    90  K.W.,   or  full  load, 

P0  =  140  K.W.,  or  J\  =  135  K.W.,   or  150%  of  load. 

Such   curves  are   called   "  Phase  Characteristics  of  the 
Synchronous  Motor." 

We  have, 

P0  =  iEp  —  rr, 
Hence, 

PO  4-  **r 


P  = 


iE 


-  irf  +  (Eg  - 


Similar  phase  characteristics  exist  also  for  the  synchron- 
ous generator,  but  are  of  less  interest.  As  seen,  on  each 
of  the  four  phase  characteristics  a  certain  field  excitation 


100  iiOO  300  400  500  600  700  800  COO  100011001200130014001500.16001700180019002000 

Fig.  59. 

gives  minimum  current,  a  lesser  excitation  gives  lagging  cur- 
rent, a  greater  excitation  leading  current.  The  higher  the 
synchronous  reactance  x#  and  thus  the  armature  reaction  of 
the  synchronous  motor,  the  flatter  are  the  phase  character- 
istics ;  that  is,  the  less  sensitive  is  the  synchronous  motor 
for  a  change  of  field  excitation  or  of  impressed  E.M.F. 


SYNCHRONOUS  MACHINES.  147 

Thus  a  relatively  high  armature  reaction  is  desirable  in 
a  synchronous  motor  to  secure  stability  ;  that  is,  indepen- 
dence of  minor  fluctuations  of  impressed  voltage  or  of  field 
excitation. 

The  theoretical  maximum  output  of  the  synchronous 
motor,  or  the  load  at  which  it  drops  out  of  step,  at  constant 
impressed  voltage  and  frequency  is,  even  with  very  high 
armature  reaction,  usually  far  beyond  the  heating  limits  of 
the  machine.  The  actual  maximum  output  depends  on  the 
drop  of  terminal  voltage  due  to  the  increase  of  current,  and 
on  the  steadiness  or  uniformity  of  the  impressed  frequency, 
thus  upon  the  individual  conditions  of  operation,  but  is  as  a 
rule  far  above  full  load. 

Hence,  by  varying  the  field  excitation  of  the  synchronous 
motor  the  current  can  be  made  leading  or  lagging  at  will, 
and  the  synchronous  motor  thus  offers  the  simplest  means  of 
producing  out  of  phase  or  wattless  currents  for  controlling 
the  voltage  in  transmission  lines,  compensating  for  wattless 
currents  of  induction  motors,  etc.  Synchronous  machines, 
used  merely  for  supph  ing  wattless  currents,  that  is,  synchro- 
nous motors  or  generators  running  light,  with  over  excited 
or  under  excited  field,  are  called  synchronous  compensators. 
They  can  be  used  as  exciters  for  induction  generators,  as  com- 
pensators for  the  wattless  lagging  currents  of  induction 
motors,  etc.  Sometimes  they  are  called  "  rotary  condensers  " 
or  dynamic  condensers  when  used  only  for  producing  leading 
currents. 

IX.     Magnetic  Characteristic  or  Saturation  Curve. 

The  dependence  of  the  induced  E.M.F.,  or  terminal  volt- 
age at  open  circuit  upon  the  field  excitation,  is  called  the 
"Magnetic  Characteristic,"  or  "Saturation  Curve,"  of  the 
synchronous  machine.  It  has  the  same  general  shape  as 
the  curve  of  magnetic  induction,  consisting  of  a  straight  part 
below  saturation,  a  bend  or  knee,  and  a  saturated  part  beyond 
the  knee.  Generally  the  change  from  the  unsaturated  to  the 


148  ELECTRICAL   ENGINEERING 

over  saturated  portion  of  the  curve  is  more  gradual,  thus 
the  knee  is  less  pronounced  in  tne  magnetic  characteristic  of 
the  synchronous  machines,  since  the  different  parts  of  the 
magnetic  circuit  approach  saturation  successively. 

The  dependence,  of  the  terminal  voltage  upon  the  field 
excitation,  at  constant  full-load  current  flowing  through  the 
armature  into  a  non-inductive  circuit,  is  called  the  "  Load 
Saturation  Curve "  of  the  synchronous  machine.  It  is  a 
curve  approximately  parallel  to  the  no-load  saturation  curve, 
but  starting  at  a  definite  value  of  field  excitation  for  zero 
terminal  voltage,  the  field  excitation  required  to  send  full- 
load  current  through  the  armature  against  its  synchronous 
impedance. 

The  ratio,  t^_  ^   jF 

~dE  ''  ~E 

is  called  the  "  Saturation  Coefficient "  of  the  machine.  It 
gives  the  ratio  of  the  proportional  change  of  field  excitation 
required  for  a  change  of  voltage. 

In  Fig.  60  is  shown  the  magnetic  characteristic  or 
no-load  saturation  curve  of  a  synchronous  generator,  the 
load  saturation  curve  and  the  no-load  saturation  coefficient, 
assuming  E  =  1000,  /  =  100  as  full-load  values. 

In  the  preceding  the  characteristic  curves  of  synchron- 
ous machines  were  discussed  under  the  assumption  that  the 
saturation  curve  is  a  straight  line  ;  that  is,  the  synchronous 
machines  working  below  saturation. 

The  effect  of  saturation  on  the  characteristic  curves  of 
synchronous  machines  is  as  follows :  The  compounding 
curve  is  impaired  by  saturation.  That  is,  a  greater  change 
of  field  excitation  is  required  with  changes  of  load.  Under 
load  the  magnetic  density  in  the  armature  corresponds  to  the 
true  induced  E.M.F.  Ev  the  magnetic  density  in  the  field 
to  the  virtual  induced  E«M^F.  Ev  Both,  especially  the  latter, 
are  higher  than  the  no-Load  E,M,F.  or  terminal  voltage  E  in 
the  generator,  and  thus  a  greater  increase  of  field  excitation 
is  required  in  presence  of  saturation  than  in  the  absence 


S  YNCIIK  ONO  US  MA  CHINES. 


149 


thereof.  In  addition  thereto,  due  to  the  counter  M.M.F.  of 
the  armature  current,  the  magnetic  stray  field,  that  is,  that 
magnetic  flux  which  leaks  from  field  pole  to  field  pole 
through  the  air,  increases  under  load,  especially  with  induc- 
tive load  where  the  armature  M.M.F.  .directly  opposes  the 


1 

SYI 

JCHRONOU'S  GENERATOR. 
SATURATION  CURVES. 

L 

1500 

.  

== 

^—  —  • 
—  ' 

—  

400 

^ 

^ 

^^**~^ 

1300 

/ 

/ 

/ 

/* 

1200 

/ 

* 

- 

I 

1100 

/ 

'/\ 

J 

1 

1000 

,1 

/  3 
'_$ 

y 

. 

ff 

900 

t 

I 

/ 

/, 

4 

/ 

800 

$ 

I 

I 

700 

/ 

/ 

600 

/ 

/ 

/ 

/ 

500 

/ 

/ 

/ 

/ 

400 

/ 

I 

/ 

/ 

300 

/ 

/ 

/ 

2OO 

/ 

fr 

100 

/ 

10 

DO 

20 

00 

30 

5O 

4C 

00 

5f 

oo 

60 

00 

70 

00 

O 

Fig.  60. 

field,  and  thus  a  still  further  increase  of  density  is  required 
in  the  field  magnetic  circuit  under  load.  In  consequence 
thereof,  at  high  saturation  the  load  saturation  curve  differs 
more  from  the  no-load  saturation  curve  than  corresponds 
to  the  synchronous  inpedance  of  the  machine. 

The  regulation  becomes  better  by  saturation  ;  that  is, 
the   increase  of  voltage  from  full  load  to  no  load  at  con- 


150  ELECTRICAL   ENGINEERING. 

stant  field  excitation  is  reduced,  the  voltage  being  limited 
by  saturation.  Owing  to  the  greater  difference  of  field  ex- 
citation between  no  load  and  full  load  in  the  case  of  magnetic 
saturation,  the  improvement  in  regulation  is  somewhat 
lessened  however,  or  may  in  some  circumstances  be  lost 
altogether. 

X.    Efficiency  and  Losses. 

Besides  these  curves  the  efficiency  curves  are  of  interest. 
The  efficiency  of  alternators  and  synchronous  motors  is 
usually  so  high  that  a  direct  determination  by  measuring 
the  mechanical  power  and  the  electric  power  is  less  reliable 
than  the  method  of  adding  the  losses,  and  the  latter  is 
therefore  commonly  used. 

The  losses  consist  of, 

Resistance  loss  in  the  armature  or  the  induced  member, 

Resistance  loss  in  the  field, 

Hysteresis  and  eddy  current  losses  in  the  magnetic  circuit, 

Friction  and  windage  losses, 

And  eventually  load  losses,  that  is,  eddy  currents  or  hyste- 
resis due  to  the  current  flowing  in  the  armature  under 
load. 

The  resistance  loss  in  the  armature  is  proportional  to 
the  square  of  the  current  /. 

The  resistance  loss  in  the  field  is  proportional  to  the 
square  of  the  field  excitation,  that  is,  the  square  of  the 
nominal  induced  or  counter  E.M.F.  EQ. 

The  hysteresis  loss  is  proportional  to  the  1.6th  power  cf 
the  real  induced  E.M.F.  El  =  E±Ir. 

The  eddy  current  loss  is  usually  proportional  to  the 
square  of  the  induced  E.M.F.  Er 

The  friction  and  windage  loss  is  constant. 

The  load  losses  vary  more  or  less  proportionally  to  the 
square  of  the  current  in  the  armature,  and  should  be  small 
with  proper  design.  They  can  be  represented  by  an  "effec- 
tive "  armature  resistance. 


S  YNCJIR  ONO  US  MA  CHINES. 


151 


Assuming  in  the  preceding  instance, 

A  friction  loss  of  2000  watts, 

A  hysteresis  loss   of  3000  watts,   at    the    induced   E.M.F. 

£t =  1000 ; 

A  resistance  loss  in  the  field  of  800  watts,  at  £0  —  1000 ; 
Load  loss  at  full  load  of  GOO  watts. 

The  loss  curves  and  efficiency  curves  are  plotted  in  Fig. 
61  for  the  generator  with  the  current  output  at  non-inductive 
load  or  w  =  0  as  abscissae,  and  in  Fig.  62  for  the  synchronous 
motor,  with  the  mechanical  power  output  as  abscissae. 


Fig.  61. 


XL   Unbalancing  of  Polyphase  Synchronous  Machines. 

The  preceding  discussion  applies  to  polyphase  as  well  as 
single-phase  machines.  In  polyphase  machines  the  nominal 
induced  E.M.F.'s  or  nominal  counter  E.M.F.'s  are  neces- 
sarily the  same  in  all  phases  (or  bear  a  constant  relation  to 
each  other).  Thus  in  a  polyphase  generator,  if  the  current 
or  the  phase  relation  of  the  current  is  different  in  the 
different  branches,  the  terminal  voltage  must  become  different 


152 


ELECTRICAL   ENGINEERING. 


also,  more  or  less.  This  is  called  the  unbalancing  of  the 
polyphase  generator.  It  is  due  to  different  load  or  load  of 
different  inductance  factor  in  the  different  branches. 

Inversely  in  a  polyphase  synchronous  motor,  if  the 
terminal  voltages  of  the  different  branches  are  unequal,  due 
to  an  unbalancing  of  the  polyphase  circuit,  the  synchronous 
motor  takes  more  current  or  lagging  current  from  the  branch 
of  higher  voltage,  and  thereby  reduces  its  voltage,  and  takes 


Fig.  62. 

less  current  or  leading  current*  from  the  branch  of  lower 
voltage,  or  even  returns  current  into  this  branch,  and  thus 
raises  its  voltage.  Hence  a  synchronous  motor  tends  to 
restore  the  balance  of  an  unbalanced  polyphase  system  ; 
that  is,  it  reduces  the  unbalancing  of  a  polyphase  circuit 
caused  by  an  unequal  distribution  or  unequal  phase  relation 
of  the  load  on  the  different  branches.  To  a  less  degree 
the  induction  motor  possesses  the  same  property. 

*  Since  with  lower  impressed  voltage  the  current  is  leading,  with  higher  impressed 
voltage  lagging,  in  a  synchronous  motor. 


SYNCHRONOUS  MACHINES. 


153 


XII.   Starting  of  Synchronous  Motors. 

In  starting,  an  essential  difference  exists  between  the 
single-phase  and  the  polyphase  synchronous  motor,  in  so  far 
as  the  former  is  not  self-starting,  but  has  to  be  brought  to 
complete  synchronism,  or  in  step  with  the  generator,  by  ex- 
ternal means  before  it  can  develop  torque,  while  the  poly- 
phase synchronous  motor  starts  from  rest,  and  runs  up  to 
synchronism  with  more  or  less  torque. 

In  starting,  the  field  excitation  of  the  polyphase  syn- 
chronous motor  must  be  zero  or  very  low. 

The  starting  torque  is  due  to  the  magnetic  attraction  of 
the  armature  currents  upon  the  remanent  magnetism  left 
in  the  field  poles  by  the  currents  of  the  preceding  phase,  or 
the  eddy  currents  induced  therein. 


B       ! 

a 
Fig.  63. 

Let  Fig.  63  represent  the  magnetic  circuit  of  a  poly- 
phase synchronous  motor.  The  M.M.F.  of  the  polyphase 
armature  currents  acting  upon  the  successive  projections  or 
teeth  of  the  armature,  1,  2,  3,  etc.,  reaches  a  maximum  in 
them  successively.  That  is,  the  armature  is  the  seat  of  a 
M.M.F.  rotating  synchronously  in  the  direction  of  the  arrow 
A.  The  magnetism  induced  in  the  face  of  the  field  pole 
opposite  to  the  armature  projections  lags  behind  the  inducing 
M.M.F.  due  to  hysteresis  and  induced  currents,  and  thus  is 
still  remanent,  while  the  M.M.F.  of  the  projection  1  decreases, 
and  is  attracted  by  the  rising  M.M.F.  of  projection  2,  etc., 
or,  in  other  words,  while  the  maximum  M.M.F.  in  the  arma- 


154  ELECTRICAL   ENGINEERING. 

ture  has  a  position  a,  the  maximum  induced  magnetism  in 
the  field  pole  face  still  has  the  position  b,  and  is  thus  attracted 
towards  a,  causing  the  field  to  revolve  in  the  direction  of 
the  arrow  A  (or  with  a  stationary  field,  the  armature  to 
revolve  in  the  opposite  direction  B). 

Lamination  of  the  field  poles  reduces  the  starting  torque 
caused  by  induced  currents  in  the  field  poles,  but  increases 
that  caused  by  remanent  magnetism  or  hysteresis,  due  to 
the  higher  permeability  of  the  field  poles.  Thus  the  torque 
per  volt  ampere  input  is  approximately  the  same  in  either 
case,  but  with  laminated  poles  the  impressed  voltage  required 
in  starting  is  higher  and  the  current  lower  than  with  solid 
field  poles.  In  either  case  at  full  impressed  E.M.F.,  the  start- 
ing current  of  a  synchronous  motor  is  large,  since  in  the 
absence  of  a  counter  E.M.F.  the  total  impressed  E.M.F.  has 
to  be  consumed  by  the  impedance  of  the  armature  circuit. 
Since  the  starting  torque  of  the  synchronous  motor  is  due 
to  the  magnetic  flux  induced  by  the  alternating  armature 
currents  or  the  armature  reaction,  synchronous  motors  of 
high  armature  reaction  are  superior  in  starting  torque. 

XIII.   Parallel  Operation. 

Any  alternator  can  be  operated  in  parallel,  or  synchronized 
with  any  other  alternator.  A  single-phaser  can  be  synchro- 
nized with  one  phase  of  a  polyphaser,  or  .a  quarter-phase 
machine  operated  in  parallel  with  a  three-phase  machine  by 
synchronizing  one  phase  of  the  former  with  one  phase  of  the 
latter.  Since  alternators  in  parallel  must  be  in  step  with 
each  other  and  have  the  same  terminal  voltage,  the  condition 
of  satisfactory  parallel  operation  is  that  the  frequency  of 
the  machines  is  identically  the  same,  and  the  field  excitation 
such  as  would  give  the  same  terminal  voltage.  If  this  is 
not  the  case,  cross  currents  will  flow  between  the  alternators 
in  a  local  circuit.  That  is,  the  alternators  are  not  without 
current  at  no  load,  and  their  currents  under  load  are  not  of 


SYNCHRONOUS  MACHINES.  155 

the  same  phase  and  proportional  to  their  respective  capa- 
cities. The  cross  currents  flowing  between  alternators 
when  operated  in  parallel  can  be  wattless  currents  or  energy 
currents. 

If  the  frequencies  of  two  alternators  are  identically  the 
same,  but  the  f.ekl  excitation  not  such  as  would  give  equal 
terminal  voltage  when  operated  in  parallel,  a  local  current 
flows  between  the  two  machines  which  is  wattless  and  lead- 
ing or  magnetizing  in  the  machine  of  lower  field  excitation, 
lagging  or  demagnetizing  in  the  machine  of  higher  field 
excitation.  At  loa,d  this  wattless  current  is  superimposed 
upon  the  currents  flowing  from  the  machine  into  the  external 
circuit.  In  consequence  thereof  the  current  in  the  machine 
of  higher  field  excitation  is  lagging  behind  the  current  in  the 
external  circuit,  the  current  in  the  machine  of  lower  field 
excitation  leads  the  current  in  the  external  circuit.  The 
currents  in  the  two  machines  are  thus  out  of  phase  with  each 
other,  and  their  sum  larger  than  the  joint  current,  or  current 
in  the  external  circuit.  Since  it  is  the  armature  reaction  of 
leading  or  lagging  current  which  makes  up  the  difference 
between  the  impressed  field  excitation  and  the  field  excita- 
tion required  to  give  equal  terminal  voltage,  it  follows  that 
the  lower  the  armature  reaction,  that  is,  the  closer  the  regu- 
lation of  the  machines,  the  more  sensitive  they  are  for 
inequalities  or  variations  of  field  excitation.  Thus,  too  low 
armature  reaction  is  undesirable  for  parallel  operation. 

With  identical  machines  the  changes  in  field  excitation 
required  for  changes  of  load  must  be  the  same.  With  ma- 
chines of  different  compounding  curves  the  changes  of  field 
excitation  for  varying  load  must  be  different,  and  such  as 
correspond  to  their  respective  compounding  curves,  if  watt- 
less currents  shall  be  avoided.  With  machines  of  reason- 
able armature  reaction  the  wattless  cross  currents  are  small, 
even  with  relatively  considerable  inequality  of  field  excita- 
tion. Machines  of  high  armature  reaction  have  been  oper- 
ated in  parallel  under  circumstances  where  one  machine 


156  ELECTRICAL   ENGINEERING. 

was  entirely  without  field  excitation,  while  the  other  carried 
twice  its  normal  field  excitation,  with  wattless  currents, 
however,  of  the  same  magnitude  as  full-load  current. 

XIV.    Division  of  Load  in  Parallel  Operation. 

Much  more  important  than  equality  of  terminal  voltage 
before  synchronizing  is  equality  of  frequency.  Inequality  of 
frequency,  or  rather  a  tendency  to  inequality  of  frequency 
(since  by  necessity  the  machines  hold  each  other  in  step 
or  at  equal  frequency),  causes  cross  currents  to  flow  which 
transfer  energy  from  the  machine  whose  driving  power  tends 
to  accelerate  to  the  machine  whose  driving  power  tends  to 
slow  down;  and  thus  relieves  the  latter,  by  increasing  the  load 
on  the  former.  Thus  these  cross  currents  are  energy  cur- 
rents, and  cause  at  no  load  or  light  load  the  one  machine  to 
drive  the  other  as  synchronous  motor,  while  under  load  the 
result  is  that  the  machines  do  not  share  the  load  in  propor- 
tion to  their  respective  capacities. 

The  speed  of  the  prime  mover,  as  steam  engine  or  tur- 
bine, changes  with  the  load.  The  frequency  of  alternators 
driven  thereby  must  be  the  same  when  in  parallel.  Thus 
their  respective  loads  are  such  as  to  give  the  same  speed  of 
the  prime  mover  (or  rather  a  speed  corresponding  to  the 
same  frequency).  Hence  the  division  of  load  between  alter- 
nators connected  to  independent  prime  movers  depends 
almost  exclusively  upon  the  speed  regulation  .of  the  prime 
movers.  To  make  alternators  divide  the  load  in  propor- 
tion to  their  capacities,  the  speed  regulation  of  their  prime 
movers  must  be  the  same,  that  is,  the  engines  or  turbines 
must  drop  in  speed  from  no  load  to  full  load  by  the  same 
percentage  and  in  the  same  manner. 

If  the  regulation  of  the  prime  movers  is  not  the  same, 
the  load  is  not  divided  proportionally  between  the  alternators, 
but  the  alternator  connected  to  the  prime  mover  of  closer 
speed  regulation  takes  more  than  its  share  of  the  load  under 


SYNCHRONOUS  MACHINES.  157 

heavy  loads,  or  less  under  light  loads.  Thus,  too  close 
speed  regulation  of  prime  movers  is  not  desirable  in  parallel 
operation  of  alternators. 

XV.   Fluctuating  Cross  Currents  in  Parallel  Operation. 

In  alternators  operated  from  independent  prime  movers, 
it  is  not  sufficient  that  the  average  frequency  corresponding 
to  the  average  speed  of  the  prime  movers  is  the  same,  but 
still  more  important  that  the  frequency  is  the  same  at  any 
instant,  that  is,  that  the  frequency  (and  thus  the  speed  of  the 
prime  mover)  is  constant.  In  rotary  prime  movers,  as  tur- 
bines or  electric  motors,  this  is  usually  the  case ;  but  with 
reciprocating  machines,  as  steam  engines,  the  torque  and 
thus  the  speed  of  rotation  rises  and  falls  periodically  during 
each  revolution,  with  the  frequency  of  the  engine  impulses. 
The  alternator  connected  with  the  engine  will  thus  not  have 
uniform  frequency,  but  a  frequency  which  pulsates ;  that  is, 
rises  and  falls.  The  amplitude  of  this  pulsation  depends 
upon  the  type  of  the  engine,  and  the  momentum  of  its  fly- 
wheel, and  the  action  of  the  engine  governor. 

If  two  alternators  directly  connected  to  equal  steam 
engines  are  synchronized  so  that  the  moments  of  maximum 
frequency  coincide,  no  energy  cross  currents  flow  between 
the  machines,  but  the  frequency  of  the  whole  system  rises 
and  falls  periodically.  In  this  case  the  engines  are  said  to 
be  synchronized.  The  parallel  operation  of  the  alternators 
is  satisfactory  in  this  case  provided  that  the  pulsations  of 
engine  speeds  are  of  the  same  size  and  duration ;  but  appa- 
ratus requiring  constant  frequency  as  synchronous  motors 
and  especially  rotary  converters,  when  operated  from  such  a 
system,  will  give  a  reduced  maximum  output  due  to  periodic 
cross  currents  flowing  between  the  generators  of  fluctuating 
frequency  and  the  synchronous  motors  of  constant  frequency, 
and  in  an  extreme  case  the  voltage  of  the  whole  system  will 
be  caused  to  fluctuate  periodically.  Even  with  small  fluctu- 


158  ELECTRICAL   ENGINEERING. 

ations  of  engine  speed  the  unsteadiness  of  current  due  to 
this  source  is  noticeable  in  synchronous  motors  and  rotary 
converters. 

If  the  alternators  happen  to  be  synchronized  in  such  a 
position  that  the  moment  of  maximum  speed  of  the  one 
coincides  with  the  moment  of  minimum  speed  of  the  other, 
alternately  the  one  and  then  the  other  alternator  will  run 
ahead,  and  thus  a  pulsating  energy  cross  current  flow  between 
the  alternators,  transferring  power  from  the  leading  to  the 
lagging  machine,  that  is,  alternately  from  the  one  to  the 
other,  and  inversely,  with  the  frequency  of  the  engine  im- 
pulses. These  pulsating  cross  currents  are  the  most  undesir- 
able, since  they  tend  to  make  the  voltage  fluctuate  and  to 
tear  the  alternators  out  of  synchronism  with  each  other, 
especially  when  the  conditions  are  favorable  to  a  cumulative 
increase  of  this  effect  by  what  may  be  called  mechanical 
resonance  (hunting)  of  the  engine  governors,  etc.  They  de- 
pend upon  the  synchronous  impedance  of  the  alternators 
and  upon  their  phase  difference,  that  is,  the  number  of  poles 
and  the  fluctuation  of  speed,  and  are  specially  objectionable 
when  operating  synchronous  apparatus  in  the  system. 

Thus,  for  instance,  if  two  80-polar  alternators  are  directly 
connected  to  single  cylinder  engines  of  1%  speed  variation 
per  revolution,  twice  during  each  revolution  the  speed  will 
rise,  twice  fall  ;  and  consequently  the  speed  of  each  alter- 
nator will  be  above  average  speed  during  a  quarter  revolu- 
tion. Since  the  maximum  speed  is  J%  above  average,  the 
mean  speed  during  the  quarter  revolution  of  high  speed  is 
1%  above  average  speed,  and  by  passing  over  20  poles  the 
armature  of  the  machine  will  during  this  time  run  ahead  of 
its  mean  position  by  \%  of  20  or  ^  pole,  that  is  y\f  =  9°. 
If  the  armature  of  the  other  alternator  at  this  moment  is 
behind  its  average  position  by  9°,  the  phase  displacement 
between  the  alternator  E.M.F.'s  is  18°  ;  that  is,  the  alter- 
nator E.M.F.'s  are  represented  by  OEl  and  OE2  in  Fig.  64, 
and  when  running  in  parallel  the  E.M.F.  OR '=  E^EZ  is 


S  YNCIIR  ONO  US  MA  CHINES. 


159 


short  circuited  through  the  synchronous  impedance  of  the 
two  alternators. 

Since  Ef  =  O£1=  2  £1  sin  9°,  the  maximum  cross  cur- 
rent is 

7-1         •        f\r>  -4   **  f*      r* 

,      £l  sin  9        .lob  El          „ 

/'     _  L *       I      'V*A         / 

J.         -  -  —     .-LtJU    JQ, 

^0  Z0 

where  70=  —  =  short    circuit  current  of  alternator  at  full- 

5° 

load  excitation.  Thus  if  the  short  circuit  current  of  the 
alternator  is  only  twice  full-load  current,  the  cross  current 
is  31.2%  of  full-load  cur- 
rent. If  the  short  circuit 
current  is  6  times  full-load 
current,  the  cross  current  0 
is  96.3%  of  full-load  cur- 
rent or  practically  equal 
to  full-load  current.  Thus 
the  smaller  the  armature 
reaction  or  the  better  the 
regulation,  the  larger  are 
the  pulsating  cross  currents  flowing  between  the  alternators, 
due  to  inequality  of  the  rate  of  rotation  of  the  prime  movers. 
Hence  for  satisfactory  parallel  operation  of  alternators  con- 
nected to  steam  engines,  a  certain  amount  of  armature  re- 
action is  desirable  and  very  close  regulation  undesirable. 

By  the  transfer  of  energy  between  the  machines  the 
pulsations  of  frequency,  and  thus  the  cross  currents,  are 
reduced  somewhat.  Very  high  amature  reaction  is  objec- 
tionable also,  since  it  reduces  the  synchronizing  power,  that 
is,  the  tendency  of  the  machines  to  hold  each  other  in  step, 
by  reducing  the  energy  transfer  between  the  machines. 

As  seen  herefrom,  the  problem  of  parallel  operation  of 
alternators  is  almost  entirely  a  problem  of  the  regulation  of 
their  prime  movers,  especially  steam  engines,  but  no  electrical 
problem  at  all. 

From  Fig.  64  it  is  seen  that  the  E.M.F.  OE'  or 


Fig.  64-. 


160  ELECTRICAL   ENGINEERING. 

which  causes  the  cross  current  between  two  alternators  in 
parallel  connection,  if  their  E.M.F.'s  OE^  and  OEZ  are  out 
of  phase,  is  approximately  in  quadrature  with  the  E.M.F.'s 
OEl  and  OE2  of  the  machines,  if  these  latter  two  E.M.F.'s 
are  equal  to  each  other.  The  cross  current  between  the 
machines  lags  behind  the  E.M.F.  producing  it,  OE',  by  the 

'V- 

angle  w,  where  tan  *»*=.—>  and  XQ  =  synchronous  reactance,. 

ro 

rQ  =  effective  resistance  of  alternator  armature.  The  energy 
component  of  this  cross  current,  or  component  in  phase  with 
OE' ',  is  thus  in  quadrature  with  the  machine  voltages  OEl 
and  OEV  that  is,  transfers  no  power  between  them  but  the 
power  transfer  or  equalization  of  load  between  the  two- 
machines  takes  place  by  the  wattless  or  reactive  component 
of  cross  current,  that  is,  the  component  which  is  in  quadra- 
ture with  OE ',  and  thus  in  phase  with  one,  and  in  opposition 
with  the  other  of  the  machine  E.M.F.'s  OE^  and  OEy 

Hence,  machines  without  reactance  would  have  no  syn- 
chronizing power,  or  could  not  be  operated  in  parallel.  The 
theoretical  maximum  synchronizing  power  exists  if  the  syn- 
chronous reactance  equals  the  resistance :  ,r0  =  r0.  This; 
condition,  however,  cannot  be  realized,  and  if  realized  would 
give  a  dangerously  high  synchronizing  power  and  cross  cur- 
rent. In  practice,  ;r0  is  always  very  much  greater  than  rQ> 
and  the  cross  current  thus  practically  in  quadrature  with 
OE't  that  is,  in  phase  (or  opposition)  with  the  machine  volt- 
ages, and  is  consequently  an  energy-transfer  current. 

If,  however,  the  machine  voltages  OEl  and  O£2  are  dif- 
ferent in  value  but  approximately  in  phase  with  each  other,, 
the  voltage  causing  cross  currents,  E^EV  is  in  phase  with 
the  machine  voltages  and  the  cross  currents  thus  in  quadra- 
ture with  the  machine  voltages  OE1  and  OE&  and  hence  do 
not  transfer  energy,  but  are  wattless.  In  one  machine  the 
cross  current  is  a  lagging  or  demagnetizing,  and  in  the 
other  a  leading  or  magnetizing,  current. 

Hence  two  kinds  of  cross  currents  may  exist  in  parallel 


SYNCHRONOUS  MACHINES.  161 

operation  of  alternators, —  currents  transferring  power  be- 
tween the  machines,  due  to  phase  displacement  between 
their  E.M.F.'s,  and  wattless  currents  transferring  magne- 
tization between  the  machines,  due  to  a  difference  of  their 
induced  E.M.F.'s. 

In  compound-wound  alternators,  that  is,  alternators  in 
which  the  field  excitation  is  increased  with  the  load  by 
means  of  a  series  field  excited  by  the  rectified  alternating 
current,  it  is  almost,  but  not  quite,  as  necessary  as  in  direct 
current  machines,  when  operating  in  parallel,  to  connect  all 
the  series  fields  in  parallel  by  equalizers  of  negligible  resist- 
ance, for  the  same  reason,  —  to  ensure  proper  division  of 
current  between  machines. 


162 


EL  E  C  TRIG  A  L   ENGINEERING. 


B.    COMMUTATING    MACHINES. 
I.   General. 

COMMUTATING  machines  are  characterized  by  the  com- 
bination of  a  continuously  excited  magnet  field  with  a  closed 
circuit  armature  connected  to  a  segmental  commutator. 
According  to  their  use,  they  can  be  divided  into  direct 
current  generators  which  transform  mechanical  power  into 
electric  power,  direct  current  motors  which  transform  elec- 
tric power  into  mechanical  power,  and  direct  current  con- 
verters which  transform  electric  power  into  a  different 
form  of  electric  power.  Since  the  most  important  class  of 
the  latter  are  the  synchronous  converters,  which  combine 
features  of  the  synchronous  machines  with  those  of  the  corn- 
mutating  machines,  they  shall  be  treated  in  a  separate  chapter. 
By  the  excitation  of  their  magnet  fields,  commutating 
machines  are  divided  into  magneto  machines,  in  which  the 

field  consists  of  permanent 
magnets;  separately  excited 
machines  ;  shunt  machines, 
in  which  the  field  is  ex- 
cited by  an  electric  circuit 
shunted  across  the  machine 
terminals,  and  thus  receives 
a  small  branch  current  at 
full  machine  voltage,  as 
shown  diagrammatically  in 
Fig.  65  ;  series  machines, 

Fig.  65.  •       /»    t  J 

in  which  the  electric  field 

circuit  is  connected  in  series  to  the  armature,  and  thus  re- 
ceives the  full  machine  current  at  small  voltage  (Fig. 


SHUNT  MACHINE 


COMMUTA  TING   MACHINES. 


163 


and  compound  machines  excited  by  a  combination  of  shunt 
and  series  field  (Fig.  67).  In  compound  machines  the  two 
windings  can  magnetize  either  in  the  same  direction  (cumu- 
lative compounding),  or  in  opposite  directions  (differential 


COMPOUND  MACHINE 


SERIES  MACHINE 


Fig.  66.  Fig.  67. 

compounding).  Differential  compounding  has  been  used  for 
constant-speed  motors.  Magneto  machines  are  used  only 
for  very  small  sizes. 

By  the  number  of  poles  commutating  machines  are 
divided  into  bipolar  and  multipolar  machines.  Bipolar  ma- 
chines are  used  only  in  small  sizes.  By  the  construction  of 
the  armature,  commutating  machines  are  divided  in  smooth- 
core  machines,  and  iron-clad  or  "toothed "  armature  ma- 
chines. In  the  smooth-core  machine  the  armature  winding 
is  arranged  on  the  surface  of  a  laminated  iron  core.  In  the 
iron-clad  machine  the  armature  winding  is  sunk  into  slots. 
The  iron-clad  type  has  the  advantage  of  greater  mechanical 
strength,  but  the  disadvantage  of  higher  self-induction  in 
commutation,  and  thus  requires  high  resistance,  carbon  cr 
graphite  commutator  brushes.  The  iron-clad  type  has  the 
advantage  of  lesser  magnetic  stray  field,  due  to  the  shorter 
gap  between  field  pole  and  armature  iron,  and  of  lesser  mag- 
net distortion  under  load,  and  thus  can  with  carbon  brushes 
be  operated  with  constant  position  of  brushes  at  all  loads. 
In  consequence  thereof,  for  large  multipolar  machines  the 


164 


ELECTRICAL   ENGINEERING. 


iron-clad  type  of  armature  is  generally  used,  and  the  smooth- 
core  ring  armature  type  only  for  large  multipolar,  low-voltage 
machines,  of  the  face  commutator  type,  to  secure  high 
economy  of  space. 

Either  of  these  types  can  be  drum  wound  or  ring  wound. 
The  drum  winding  has  the  advantage  of  lesser  self-induction, 
and  lesser  distortion  of  the  magnetic  field,  and  is  generally 
less  difficult  to  construct  and  thus  mostly  preferred,  except 
in  special  cases  as  to  secure  high  economy  in  space  in  low- 
voltage  multipolar  machines.  By  the  armature  winding,  the 
commutating  machines  are  divided  into  multiple  wound  and 
series  wound  machines.  The  difference  between  multiple 
and  series  armature  winding,  and  their  modifications,  can 
best  be  shown  by  diagram. 


MUCrfPLE  RING. 
Fig.  68. 


COMMUTA  TING   AT  A  CHINES. 


165 


II.     Armature  Winding. 

Fig.  68  shows  a  six-polar  multiple  ring  winding,  and  Fig. 
69  a  six-polar  multiple  drum  winding.  As  seen,  the  armature 
coils  are  connected  progressively  all  around  the  armature  in 


MULTIPLE  DRUM, FULL  PITCH. 
Fig.  69. 

closed  circuit,  and  the  connections  between  adjacent  armature 
coils  lead  to  the  commutator.  Such  an  armature  winding 
has  as  many  circuits  in  multiple,  and  requires  as  many  sets  of 
commutator  brushes,  as  poles.  Thirty-six  coils  are  shown  in 
Figs.  68  and  69,  connected  to  36  commutator  segments,  and 
the  two  sides  of  each  coil  distinguished  by  drawn  and  dotted 
lines.  In  a  drum-wound  machine,  usually  the  one  side  of  all 


166 


ELECTRICAL   ENGINEERING. 


coils  forms  the  upper  and  the  other  side  the  lower  layer  of 
the  armature  winding. 

Fig.  70  shows  a  six-polar  series  drum  winding,  with  36 
slots  and  36  commutator  segments.     In  the  series  winding 


SERIES  DRUM  WITH  CROSS-CONNECTED  COMMUTATOR. 
Fig.  70. 

the  circuit  passes  from  one  armature  coil,  not  to  the  next 
adjacent  armature  coil  as  in  the  multiple  winding,  but  first 
through  all  the  armature  coils  having  the  same  relative 
position  with  regard  to  the  magnet  poles  of  the  same  polarity, 
and  then  to  the  armature  coil  next  adjacent  to  the  first  coil. 
That  is,  all  armature  coils  having  the  same  or  approximately 
the  same  relative  position  to  poles  of  equal  polarity,  form 
one  set  of  integral  coils.  Thus  the  series  winding  has  only 


C O MM U TAXING  MACHINES. 


167 


two  circuits  in  multiple,  and  requires  two  sets  of  brushes 
only,  but  can  be  operated  also  with  as  many  sets  of  brushes 
as  poles,  or  any  intermediate  number  of  sets  of  brushes.  In 
Fig.  70,  a  series  winding  in  which  the  number  of  armature 
coils  is  divisible  by  the  number  of  poles,  the  commutator 
segments  have  to  be  cross  connected. 

This  form   of  series  winding  is  hardly  ever  used.     The 
usual  form  of  series  winding  is  the  winding  shown  by  Fig.  71. 


This  figure  shows  a  six-polar  armature  having  35  coils  and 
35  commutator  segments.  In  consequence  thereof,  the 
armature  coils  under  corresponding  poles  which  are  con- 
nected in  series  are  slightly  displaced  from  each  other,  so 


168 


ELECTRICAL   ENGINEERING. 


that  after  passing  around  all  corresponding  poles,  the  wind- 
ing leads  symmetrically  into  the  coil  adjacent  to  the  first 
armature  coil.  Hereby  the  necessity  of  commutator  cross 
connections  is  avoided,  and  the  winding  is  perfectly  sym- 
metrical. With  this  form  of  series  winding,  which  is  mostly 
used,  the  number  of  armature  coils  must  be  chosen  to  follow 


OPEN  CIRCUIT,   THREE  PHASE,  SERIES  DRUtf. 
Fig.  72. 

certain  rules.     Generally  the  number  of  coils  is  one  less  or 
one  more  than  a  multiple  of  the  number  of  poles. 

All  these  windings  are  closed  circuit  windings  ;  that  is, 
starting  at  any  point,  and  following  the  armature  conductor, 
the  circuit  returns  into  itself  after  passing  all  E.M.F.'s  twice 
in  opposite  direction  (thereby  avoiding  short  circuit).  An 
instance  of  an  open-coil  winding  is  shown  in  Fig.  72,  a  series 
connected  three-phase  Y  winding  similar  to  that  used  in  the 


CO MMUTA TING   MACHINES.  169 

Thomson-Houston  arc  machine.  Such  open-coil  windings, 
however,  cannot  be  used  in  commutating  machines.  They 
are  generally  preferred  in  synchronous  and  in  induction 
machines. 

By  leaving  space  between  adjacent  coils  of  these  wind- 
ings, a  second  winding  can  be  laid  in  between.  The  second 
winding  can  either  be  entirely  independent  from  the  first 


MULTIPLE,    DOUBLE    SPIRAL   R.LNG. 
Fig.  73. 

winding  ;  that  is,  each  of  the  two  windings  closed  upon 
itself,  or  after  passing  through  the  first  winding  the  circuit 
enters  the  second  winding,  and  after  passing  through  the 
second  winding  it  reenters  the  first  winding.  In  the  first 
case,  the  winding  is  called  a  double  spiral  winding  (or  mul- 
tiple spiral  winding  if  more  than  two  windings  are  used),  in 
the  latter  case  a  double  re-entrant  winding  (or  multiple  re- 
entrant winding).  In  the  double  spiral  winding,  the  number 
of  coils  must  be  even  ;  in  the  double  re-entrant  winding,  odd. 


170  ELECTRICAL   ENGINEERING. 

Multiple  spiral  and  multiple  reentrant  windings  can  be 
either  multiple  or  series  wound ;  that  is,  each  spiral  can  con- 
sist either  of  a  multiple  or  of  a  series  winding.  Fig.  73 
shows  a  double  spiral  multiple  ring  winding,  Fig.  74  a  double 
spiral  multiple  drum  winding,  Fig.  75  a  double  reentrant 


MULTIPLE,   DOUBLE  SPIRAL  DRUM,  FULL  PITCH. 
Fig.  74. 

multiple  drum  winding.  As  seen  in  the  double  spiral  or 
double  reentrant  multiple  winding  twice  as  many  circuits  as 
poles  are  in  multiple.  Thus  such  windings  are  mostly  used 
for  large  low-voltage  machines. 

A  distinction  is  frequently  made  between  lap  winding 
and  wave  winding.  These  are,  however,  not  different  types  ; 
but  the  wave  winding  is  merely  a  constructive  modification 
of  the  series  drum  winding  with  single-turn  coil,  as  seen  by 


COMMUTATING   MACHINES.  171 

comparing  Figs.  76  and  77.  Fig.  76  shows  a  part  of  a 
series  drum  winding  developed.  Coil  Cl  and  C.2,  having  cor- 
responding positions  under  poles  of  equal  polarity,  are  joined 
in  series.  Thus  the  end  connection  ab  of  coil  Cl  connects 
by  cross  connection  be  and  dc  to  the  end  connection  de  of 
coil  C.,.  If  the  armature  coils  consist  of  a  single  turn  only, 


MULTIPLE,   DOUBLE  REENTRANT  DRUM,   FULL  PITCH. 
Fig.  75. 

as  in  Fig.  76,  and  thus  are  open  at  d  and  e,  the  end  con- 
nection and  the  cross  connection  can  be  combined  by  passing 
from  a  in  coil  C^  directly  to  c  and  from  c  directly  to  e  in 
coil  C.2.  That  is,  the  circuit  abcde  is  replaced  by  ace.  This 
has  the  effect  that  the  coils  are  apparently  open  at  one  side. 
Such  a  winding  has  been  called  a  wave  winding.  Only 
series  windings  with  a  single  turn  per  coil  can  be  arranged 


172 


ELECTRICAL   ENGINEERING. 


as  wave  windings,  while  windings  with  several  turns  per  coil 
must  necessarily  be  lap  or  coil  windings.  In  Fig.  78  is 
shown  a  series  drum  winding  with  35  coils  and  commutator 
segments,  and  a  single  turn  per  coil  arranged  as  wave 


SERIES  LAP  WINDING 


J 


WAVE  WINDING, 
Fig.  77. 

winding.     This  winding  may  be  compared  with  the  35-coil 
series  drum  winding,  in  Fig.  71. 

Drum  winding  can  be  divided  into  full  pitch  and  frac- 
tional pitch  windings.  In  the  full  pitch  winding,  the  spread 
of  the  coil  covers  the  pitch  of  one  pole  ;  that  is,  each  coil 
covers  £  of  the  armature  circumference  in  a  six-polar  ma- 


COMMUTATING  MACHINES.  173 

chine,  etc.     In  a  fractional  pitch  winding  it  covers  less  or 
more. 

Series  drum  windings  without  cross  connected  commu- 
tator in  which  thus  the  number  of  coils  is  not  divisible  by 
the  number  of  poles,  are  necessarily  always  slightly  fractional 


SERIES  'DRUM",  WAVE  WINDING. 
Fig.  78. 

pitch;  but  generally  the  expression  "fractional  pitch  winding" 
is  used  only  for  windings  in  which  the  coil  covers  one  or 
several  teeth  less  than  correspond  to  the  pole  pitch.  Thus 
the  multiple  drum  winding  in  Fig.  69  would  be  a  fractional 
pitch  winding  if  the  coils  spread  over  only  four  or  five 
teeth  instead  of  over  six.  As  £  fractional  pitch  winding  it 
is  shown  in  Fig.  79. 

Fractional  pitch  windings  have  the  advantage  of  shorter 


174 


ELECTRICAL   ENGINEERING. 


end  connections  and  less  self-induction  in  commutation, 
since  commutation  of  corresponding  coils  under  different 
poles  does  not  take  place  in  the  same,  but  in  different,  slots, 
and  the  flux  of  self-induction  in  commutation  is  thus  more 
subdivided.  Fig.  79  shows  the  multiple  drum  winding  of 


MULTIPLE,    DRUM    %   FRACTIONAL   PITCH. 
Fig.  79. 

Fig.  69  as  fractional  pitch  winding  with  five  teeth  spread,  or 
£  pitch.  During  commutation,  the  coils  abcdef  commutate 
simultaneously.  In  Fig.  69  these  coils  lie  by  two  in  the 
same  slots,  in  Fig.  79  they  lie  in  separate  slots.  Thus,  in 
the  former  case  the  flux  of  self-induction  interlinked  with  the 
commutated  coil  is  due  to  two  coils  ;  that  is,  twice  that  in 
the  latter  case.  Fractional  pitch  windings,  however,  have 
the  disadvantage  of  reducing  the  width  of  the  neutral  zone, 


COMMUTATING   MACHINES.  175 

or  zone  without  induced  E.M.F.  between  the  poles,  in  which 
commutation  takes  place,  since  the  one  side  of  the  coil 
enters  or  leaves  the  field  before  the  other.  Therefore,  in 
commutating  machines  it  is  seldom  that  a  pitch  is  used  that 
falls  short  of  full  pitch  by  more  than  one  or  two  teeth, 
while  in  induction  machines,  occasionally  as  low  a  pitch  as 
50  %  is  used,  and  |  pitch  is  frequently  employed. 

Series  windings  find  their  foremost  application  in  ma- 
chines with  small  currents,  or  small  machines  in  which  it  is 
desirable  to  have  as  few  circuits  as  possible  in  multiple,  and 
in  machines  in  which  it  is  desirable  to  use  only  two  sets  of 
brushes,  as  in  railway  motors.  In  multipolar  machines  with 
many  sets  of  brushes,  series  winding  is  liable  to  give  selective 
commutation ;  that  is,  the  current  does  not  divide  evenly 
between  the  sets  of  brushes  of  equal  polarity. 

'  Multiple  windings  are  used  for  machines  of  large  cur- 
rents, thus  generally  for  large  machines,  and  in  large  low- 
voltage  machines  the  still  greater  subdivision  of  circuits 
afforded  by  the  multiple  spiral  and  the  multiple  reentrant 
winding  is  resorted  to. 

To  resume,  then,  armature  windings  can  be  subdivided 
into : 

a).    Ring  and  drum  windings. 

£).    Closed    circuit    and    open    circuit  windings.      Only    the 

former  can  be  used  for  commutating  machines. 
c).    Multiple  and  series  windings. 
d).    Single    spiral,    multiple    spiral,   and    multiple    reentrant 

windings.     Either  of  these  can  be  multiple  or  series 

windings. 
e).    Full  pitch  and  fractional  pitch  windings. 

III.   Induced  E.M.F.'S. 

The  formula  of  induction  of  a  direct  current  machine, 
as  discussed  in  the  preceding,  is  : 

e  = 


OF  THE 

UNIVERSITY 


176  ELECTRICAL   ENGINEERING. 

Where 

e  =  induced  E.M.F. 
N  =  frequency  =  number  of  pairs  of   poles  X  revolutions 

per  second,  in  hundreds. 

n  =  number  of  turns  in  series  between  brushes. 
<E>  ==  magnetic  flux  passing  through  the  armature  per  pole, 
in  megalines. 

In  ring-wound  machines,  3>  is  one-half  the  flux  per  field 
pole,  since  the  flux  divides  in  the  armature  into  two  circuits, 
and  each  armature  turn  incloses  only  half  the  flux  per  field 
pole.  In  ring-wound  armatures,  however,  each  armature  turn 
has  only  one  conductor  lying  on  the  armature  surface,  or  face 
conductor,  while  in  a  drum-wound  machine  each  turn  has 
two  face  conductors.  Thus,  with  the  same  number  of  face 
conductors  —  that  is,  the  same  armature  surface  —  the  same 
frequency,  and  the  same  flux  per  field  pole,  the  same  E.M.F. 
is  induced  in  the  ring-wound  as  in  the  drum-wound  armature. 

The  number  of  turns  in  series  between  brushes,  n,  is  one- 
half  the  total  number  of  armature  turns  in  a  series  wound 

armature,  —  the  total  number  of  armature  turns  in  a   single 

P 
spiral  multiple  wound  armature  with  /  poles.     It  is  one-ha:f 

as  many  in  a  double  spiral  or  double  reentrant,  one-third  as 
many  in  a  triple  spiral  winding,  etc. 

By  this  formula,  from  frequency,  series  turns  and  mag- 
netic flux  the  E.M.F.  is  found,  or  inversely  from  induced 
E.M.F.,.  frequency,  and  series  turns,  the  magnetic  flux  per 
field  pole  is  calculated. 


From  magnetic  flux,  and  section  and  lengths  of  the  dif- 
ferent parts  of  the  magnetic  circuit,  the  densities  and  the 
ampere  turns  required  to  produce  these  densities  are  derived, 
and  as  the  sum  of  the  ampere  turns  required  by  the  dif- 
ferent parts  of  the  magnetic  circuit,  the  total  ampere  turns 
excitation  per  field  pole  is  found,  which  is  required  for 
inducing  the  desired  E.M.F. 


COMMUTAT1NG   MACHINES.  177 

Since  the  formula  of  direct  current  induction  is  indepen- 
dent of  the  distribution  of  the  magnetic  flux,  or  its  wave 
shape,  the  total  magnetic  flux  and  thus  the  ampere  turns  re- 
quired therefor,  are  independent  also  of  the  distribution  of 
magnetic  flux  at  the  armature  surface.  The  latter  is  of 
importance,  however,  regarding  armature  reaction  and 
commutation. 

IV.   Distribution  of  Magnetic  Flux. 

The  distribution  of  magnetic  flux  in  the  air  gap  or  at  the 
armature  surface  can  be  calculated  approximately  by  assum- 
ing the  density  at  any  point  of  the  armature  surface  as  pro- 
portional to  the  M.M.F.  acting  thereon,  and  inversely 
proportional  to  the  nearest  distance  from  a  field  pole.  Thus, 
if  £F0  =  ampere  turns  acting  upon  air  gap  between  armature 
and  field  pole,  a  =  length  of  air  gap,  from  iron  to  iron, 
the  density  under  magnet  pole,  that  is,  in  the  range  BC  of 
Fig.  80,  is 


I  I 

Fig.  80. 

At  a  point  having  the  distance  x  from  the  end  of  the 
field  pole  on  the  armature  surface,  the  distance  from  the  next 
field  pole  is  d  =  V#2  +  x2,  and  the  density  thus, 


10       tf2  +  3? 

Herefrom  the  distribution  of  magnetic  flux  is  calculated 
and  plotted  in  Fig.  80,  for  a  single  pole  BC,  along  the 
armature  surface  A,  for  the  length  of  air  gap  a  =  1,  and  such 


178 


ELE  C  TRIG  A  L   ENGINEERING. 


a  M.M.F.  as  to  give  (B0  =  8000  under  the  field  pole,  that  is, 
for  JF0  =  6400  or  OC0  =  8000. 

Around  the  surface  of  the  direct  current  machine  arma- 
ture, alternate  poles  follow  each  other.  Thus  the  M.M.F. 
is  constant  only  under  each  field  pole,  but  decreases  in  the 
space  between  the  field  poles,  from  C  to  E  in  Fig.  81, 


L- 


Fig.  81. 

from  full  value  at  C  to  full  value  in  opposite  direction  at  E. 
The  point  D  midway  between  C  and  E,  at  which  the  M.M.F. 
of  the  field  equals  zero,  is  called  the  "neutral."  The  dis- 
tribution of  M.M.F.  of  field  excitation  is  thus  given  by  the 
line  $  in  Fig.  81.  The  distribution  of  magnetic  flux  as 
shown  in  Fig.  81  by  (&,  is  derived  by  the  formula, 


Where 


This  distribution  of  magnetic  flux  applies  only  to  the  no- 
load  condition.  Under  load,  that  is,  if  the  armature  carries 
current,  the  distribution  of  flux  is  changed  by  the  M.M.F.  of 
the  armature  current,  or  armature  reaction. 

Assuming  the  brushes  set  at  the  middle  points  between 
adjacent  poles,  D  and  G,  Fig.  82,  the  M.M.F.  of  the  arma- 
ture is  maximum  at  the  point  connected  with  the  commutator 
brushes,  thus  in  this  case  at  the  points  D  and  G,  and 
gradually  decreases  from  full  value  at  D  to  equal  but  op- 
posite value  at  G\  as  shown  by  the  line  /in  Fig.  82,  while 
the  line  (F0  gives  the  field  M.M.F.  or  impressed  M.M.F. 


COMMUTA  TING   MA  CHINES. 


179 


If  ;/  =  number  of  turns  in  series  between  brushes  per 
pole,  i  =  current  per  turn,  the  armature  reaction  is/=  ni 
ampere  turns.  Adding /and  0^  gives  the  resultant  M.M.F. 
JF,  and  therefrom  the  magnetic  distribution. 


~~Wd 

The  latter  is  shown  as  line  (Bi  in  Fig.  82. 

With    the    brushes  set  midway  between  adjacent  field 
poles,  the  armature  M.M.F.  is  additive  on  one  side,  and  sub- 


J 


Fig.  82. 


tractive  on  the  other  side  of  the  center  of  the  field  pole. 
Thus  the  magnetic  intensity  is  increased  on  one  side,  and 
decreased  on  the  other.  The  total  M.M.F.,  however,  and 
thus  neglecting  saturation,  the  total  flux  entering  the  arma- 
ture, are  not  changed.  Thus,  armature  reaction,  with  the 
brushes  midway  between  adjacent  field  poles,  acts  distorting 
upon  the  field,  but  neither  magnetizes  nor  demagnetizes,  if 
the  field  is  below  saturation. 

The  distortion  of  the  magnetic  field  takes  place  by  the 
armature  ampere  turns  beneath  the  pole,  or  from  B  to  C. 
Thus,  if  v  =  pole  arc,  that  is,  the  angle  covered  by  pole  face 
(two  poles  or  one  complete  period  being  denoted  by  360°), 

the  distorting  ampere  turns  armature  reaction  are^—. 

180 


180  ELECTRICAL   ENGINEERING 


As  seen,  in  the  assumed  instance,  Fig.  82,  where/  = 


3  ST. 


the  M.M.F.  at  the  two  opposite  pole  corners,  and  thus  the 
magnetic  densities,  stand  in  the  proportion  1  to  3.  As  seen, 
the  induced  E.M.F.  is  not  changed  by  the  armature  reaction, 
with  the  brushes  set  midway  between  the  field  poles,  except 
by  the  small  amount  corresponding  to  the  flux  entering  be- 
yond D  and  G,  that  is,  shifted  beyond  the  position  of 
brushes.  At  D,  however,  the  flux  still  enters  the  armature, 
depending  in  intensity  upon  the  armature  reaction  ;  and  thus 
with  considerable  armature  reaction,  the  brushes  when  set 
at  this  point  are  liable  to  spark.by  short  circuiting  an  active 
E.M.F.  Thus,  under  load,  the  brushes  are  shifted  towards 
the  following  pole  ;  that  is,  towards  the  direction  in  which 
the  zero  point  of  magnetic  induction  has  been  shifted  by  the 
armature  reaction. 

In  the  following  Fig.   83,  the  brushes  are  assumed  as 
shifted  to  the  corner  of  the  next  pole  £,  respectively  B. 


-I        G 


Fig.  83. 


In  consequence  thereof,  the  subtract ive  range  of  the  arma- 
ture M.M.F.  is  larger  than  the  additive,  and  the  resultant 
M.M.F.  SF  =  $Q  +  f  is  decreased.  That  is,  with  shifted 
brushes  the  armature  reaction  demagnetizes  the  field.  The 
demagnetizing  armature  ampere  turns  are  PM ;  that  is, 


COMMUTATING   MACHINES.  181 

GB 


f.     That  is,  if  h  —  angle  of  shift  of  brushes  or  angle 

of  lead  (=  GB  in  Fig.  83),  assuming  the  pitch  of  two  poles 
=  360°,  the  demagnetizing  component  of  armature  reaction 

Wif  vf 

is    .,  OA  ,  the  distorting  component  is  ^TTTT:  where  v  =  pole  arc. 
loU  JLoU 

Thus,  with  shifted  brushes  the  field  excitation  has  to  be 
increased  under  load  to  maintain  the  same  total  resultant 
M.M.F.  ;  that  is,  the  same  total  flux  and  induced  E.M.F. 
Hence,  in  Fig.  83  the  field  excitation  &0  has  been  assumed 

•2/if      f 
by  ^-^r  =  3  larger   than   in   the   previous   figures,   and    the 

magnetic  distribution  (&L  plotted  for  these  values. 


V.   Effect  of  Saturation  on  Magnetic  Distribution. 

The  preceding  discussion  of  Figs.  80  to  83  omits  the 
effect  of  saturation.  That  is,  the  assumption  is  made  that 
the  magnetic  material  near  the  air  gap,  as  pole  face  and 
armature  teeth,  are  so  far  below  saturation  that  at  the  de- 
magnetized pole  corner  the  magnetic  density  decreases,  at 
the  strengthened  pole  corner  increases,  proportionally  to  the 
M.M.F. 

The  distribution  of  M.M.F.  obviously  is  not  affected  by 
saturation,  but  the  distribution  of  magnetic  flux  is  greatly 
changed  thereby.  To  investigate  the  effect  of  saturation, 
in  Figs.  84  to  87  the  assumption  has  been  made  that  the  air 
gap  is  reduced  to  one-half  its  previous  value,  a  =  .5,  thus 
•consuming  only  one-half  as  many  ampere  turns,  and  the 
other  half  of  the  ampere  turns  are  consumed  by  saturation 
of  the  armature  teeth.  The  length  of  armature  teeth  is 
assumed  as  3.2,  and  the  space  filled  by  the  teeth  is  assumed 
as  consisting  of  one-third  of  iron  and  two-thirds  of  non- 
magnetic material  (armature  slots,  ventilating  ducts,  insula- 
tion between  laminations,  etc.). 

In  Figs.  84,  85,  86,  87,  curves  are  plotted  corresponding 


182 


ELECTRICAL   ENGINEERING. 


Fig.  84. 


J 


L 


J 


Fig.  85. 


L 


Fig.  87. 


COMMUTATIA7G   MACHINES.  183 

to  those  in  Figs.  80,  81,  82,  and  83.  As  seen,  the  spread 
of  magnetic  induction  at  the  pole  corners  is  greatly  increased, 
but  farther  away  from  the  field  poles  the  magnetic  distri- 
bution is  not  changed. 

The  magnetizing,  or  rather  demagnetizing,  effect  of  the 
load  with  shifted  brushes  is  not  changed.  The  distorting 
effect  of  the  lead  is,  however,  very  greatly  decreased,  to  a 
small  percentage  of  its  previous  value,  and  the  magnetic 
field  under  the  field  pole  is  very  nearly  uniform  under  load. 

The  reason  is:  Even  a  very  large  increase  of  M.M.F. 
does  not  much  increase  the  density,  the  ampere  turns  being 
consumed  by  saturation  of  the  iron,  ar.d  even  with  a  large 
decrease  of  M.M.F.,  the  density  is  not  decreased  much,  since 
with  a  small  decrease  of  density  the  ampere  turns  corsumed 
by  the  saturation  of  the  iron  become  available  for  the  gap. 

Thus,  while  in  Fig.  83  the  densities  at  the  center  and 
the  two  pole  corners  of  the  field  pole  are  8000,  12000,  and 
4000,  with  the  saturated  structure  in  Fig.  87,  they  are 
8000,  9040,  and  6550. 

At  or  near  the  theoretical  neutral,  however,  the  satura- 
tion has  no  effect. 

That  is,  saturation  of  the  armature  teeth  affords  a  means 
of  reducing  the  distortion  of  the  magnetic  field,  or  the 
shifting  of  flux  at  the  pole  corners,  and  is  thus  advantageous 
for  machines  which  shall  operate  over  a  wide  range  of  lead 
with  fixed  position  of  brushes  if  the  brushes  are  shifted  to 
near  the  next  following  pole  corner. 

It  offers  no  direct  advantage,  however,  for  machines 
commutating  with  the  brushes,  midway  between  the  field 
poles,  as  converters. 

A  similar  effect  to  saturation  in  the  armature  teeth,  is 
produced  by  saturation  of  the  field  pole  face,  or  mere  par- 
ticularly, saturation  of  the  pole  corners  of  the  field. 


184  ELECTRICAL   ENGINEERING. 

VI.   Effect  of  Slots  on  Magnetic  Flux. 

With  slotted  armatures  the  pole  face  density  opposite 
the  armature  slots  is  less  than  that  opposite  the  armature 
teeth,  due  to  the  greater  distance  of  the  air  path  in  the 
former  case.  Thus,  with  the  passage  of  the  armature  slots 
across  the  field  pole  a  local  pulsation  of  the  magnetic  flux 
in  the  pole  face  is  produced,  which,  while  harmless  with 
laminated  field  pole  faces,  induces  eddy  currents  in  solid  pole 
pieces.  The  frequency  of  this  pulsation  is  extremely  high, 
and  thus  the  energy  loss  due  to  eddy  currents  in  the  pole 
faces  may  be  considerable,  even  with  pulsations  of  small 
amplitude.  If  s  =  peripheral  speed  of  the  armature  in 
centimeters  per  second,  /  =  pitch  of  armature  slot  (that  is, 
width  of  one  slot  and  one  tooth  at  armature  surface),  the 
frequency  is  JVt  =  s  /p.  Or,  if  N  =  frequency  of  machine, 
q.  =  number  of  armature  slots  per  pair  of  poles,  N^  =  qN. 

_^          For  instance,  N=  33.3, 
_$'£=:  51,  thus  NI=  1TOO. 

Under  the  assumption, 
width  of  slots  equals  width 
of  teeth  =  2  x  width  of  air 
gap,  the  distribution  of 
magnetic  flux  at  the  pole 
face  is  plotted  in  Fig.  88. 

The  drop  of  density 
opposite  each  slot  consists 
of  two  curved  branches 
equal  to  those  in  Fig.  80, 


r 


that  is,  calculated  by 


Fig.  88. 


The  average  flux  is  7525.  That  is,  by  cutting  half  the 
armature  surface  away  by  slots  of  a  width  equal  to  twice  the 
length  of  air  gap,  the  total  flux  under  the  field  pole  is  re- 
duced only  in  the  proportion  8000  to  7525,  or  about  6%. 


COMMUTATING  MACHINES.  185 

The  flux,  (B  pulsating  between  8000  and  5700,  is  equiva- 
lent to  a  uniform  flux  (Bx=75 25  superposed  with  an  alter- 
nating flux  (B0,  shown  in  Fig.  89,  with  a  maximum  of  475 
and  a  minimum  of  1825.  This  alternating  flux  (B0  can,  as 


r\     r\ 


Fig.  89. 

regards  induction  of  eddy  currents,  be  replaced  by  the  equiva- 
lent sine  wave  (B^,  that  is,  a  sine  wave  having  the  same 
effective  value  (or  square  root  of  mean  square).  The  effec- 
tive value  is  718. 

The  pulsation  of  magnetic  flux  farther  in  the  interior 
of  the  field-pole  face  can  be  approximated  by  drawing  curves 
equi-distant  from  (B0.  Thus  the  curves  (B^  (&v  (B15,  (B2,  (B25, 
and  (E3,  are  drawn  equidistant  from  (B0  in  the  relative  dis- 
tances .5,  1,  1.5,  2,  2.5,  and  3  (where  a  =  1  is  the  length  of 
air  gap).  They  give  the  effective  values  : 

«0  ®.6  &!  (Bj.5  (B2  (B2.5  (B3 

718         373         184         119         91         69          57 
That  is,  the  pulsation  of  magnetic  flux  rapidly  disappears 
towards  the    interior    of   the    magnet    pole,  and  still  more 
rapidly  the  energy  loss  by  eddy  currents,  which  is  propor- 
tional to  the  square  of  the  magnetic  density. 

In  calculating  the  effect  of  eddy  currents,  the  magne- 
tizing effect  of  eddy  currents  may  be  neglected  (which  tends 
to  reduce  the  pulsation  of  magnetism) ;  this  gives  the  upper 
limit  of  loss. 

Let  (B  =  effective  density  of  the  alternating  magnetic  flux, 
s  =  peripheral   speed  of    armature   in   centimeters  per 

second,  and 
/  =  length  of  pole  face  along  armature. 


186 


ELE  C  TRIG  A  L    ENGINEERING. 


The  E.M.F.  induced  in  the  pole  face  is  then, 
e  =  sl<$>  X  1C-8, 

and  the  current  flowing   in  a  strip  of    thickness  &  and  one 
centimeter  width, 


where, 

p  —  resistivity  of  the  material. 

Thus  the  effect  of  eddy  currents  in  this  strip  is, 


or  per  cm3, 


. 
A/  =  <?A/  =  - 


That  is,  proportional  to  the  square  of  the  effective  value 
of  magnetic  pulsation,  the  square  cf  peripheral  speed,  and 
inversely  proportional  to  the  resistivity. 

Thus,  assuming  for  instance, 

j  =  2,000, 

p  =    20  x  10  ^6,  for  cast  steel, 

p  =  100  x  10~6,  for  cast  iron. 

we  have  in  the  above  given  instance, 


AT  DISTANCE 

/T> 

P 

FROM 
POLEFACE. 

OD. 

CAST  STEEL. 

CAST  IKON. 

0 

718 

10.3 

2.06 

a 

373 

2.78 

.56 

2 

a 

184 

.677 

.135 

3  a 

119 

.283 

.057 

2 

2a 

91 

.166 

.033 

Ha 

69 

.095 

.019 

2 

Ba 

"57 

.065 

.013 

COMMUTATING   MACHINES.  187 


VII.   Armature  Reaction. 

At  no  load,  that  is,  with  no  current  flowing  through  the 
armature  or  induced  circuit,  the  magnetic  field  of  the  corn- 
mutating  machine  is  symmetrical  with  regard  to  the  field 
poles. 

Thus  the  density  at  the  armature  surface  is  zero  at  the 
point  or  in  the  range  midway  between  adjacent  field  poles. 
This  point,  or  range,  is  called  the  "  neutral "  point  or 
"neutral"  range  of  the  commutating  machine. 

Under  load  the  armature  current  represents  a  M.M.F. 
acting  in  the  direction  from  commutator  brush  to  commu- 
tator brush  of  opposite  polarity,  that  is,  in  quadrature  with 
the  field  M.M.F.  if  the  brushes  stand  midway  between  the 
field  poles ;  or  shifted  against  the  quadrature  position  by  the 
same  angle  by  which  the  commutator  brushes  are  shifted, 
which  angle  is  called  the  angle  of  lead. 

If  ;/  =  turns  in  series  between  brushes  per  pole,  and  i  = 
current  per  turn,  the  M.M.F.  of  the  armature  is  f=  ni  per 
pole.  Or,  if  m  —  total  number  of  turns  on  the  armature, 
b  =  number  of  turns  or  circuits  in  multiple,  2/  =  numbers 
of  poles,  and  z'0=  total  armature  current,  the  M.M.F.  of  the 

armature  per  pole  is  f  =- — -.     This  M.M.F.  is  called  the 

armature  reaction  of  the  continuous  current  machine. 

Since  the  armature  turns  are  distributed  over  the  total 
pitch  of  pole,  that  is,  a  space  of  the  armature  surface  repre- 
senting 180°,  the  resultant  armature  reaction  is  found  by 

(  4-  90      2 
multiplying  f  with  the  average  cos  5  =  - ,  and  is  thus, 

(    —  90  7T 

/.-¥-" 

When  comparing  the  armature  reaction  of  commutating 
machines  with  other  types  of  machines,  as  synchronous 

2/ 

machines,  etc.,  the  resultant  armature  reaction  ff*—   has 


188  ELECTRICAL   ENGINEERING. 

to  be  used.  In  discussing  commutating  machines  proper, 
however,  the  value  /=  ni  is  usually  considered  as  the  arma- 
ture reactkJfi. 

The  armature  reaction  of  the  commutating  machine  has 
a  distorting  and  a  magnetizing  or  demagnetizing  action  upon 
the  magnetic  field.  The  armature  ampere  turns  beneath  the 
field  poles  have  a  distorting  action  as  discussed  under 
"  Magnetic  Distribution  "  in  the  preceding  paragraph.  The 
armature  ampere  turns  between  the  field  poles  have  no  effect 
upon  the  resultant  field  if  the  brushes  stand  at  the  neutral  ; 
but  if  the  brushes  are  shifted,  the  armature  ampere  turns 
inclosed  by  twice  the  angle  of  lead  of  the  brushes,  have  a 
demagnetizing  action. 

Thus,  if  v  =  pole  arc  as  fraction  of  pole  pitch,  and  h  = 
shift  of  brushes  as  fraction  of  pole  pitch,  /the  M.M.F.  of 
armature  reaction,  and  SF0  the  M.M.F.  of  field  excitation  per 
pole,  the  demagnetizing  component  of  armature  reaction  is 
2/if,  the  distorting  component  of  armature  reaction  is  if,  and 

the  magnetic  density  at  the  strengthened  pole  corner  thus 

vf 
corresponds  to  the  M.M.F.  3r0+-Q-,  at  the  weakened  field 

corner  to  the  M.M.F.  &  —  -> 


VIII.   Saturation  Curves. 

As  saturation  curve  or  magnetic  characteristic  of  the 
commutating  machine  is  understood  the  curve  giving  the 
induced  voltage,  or  terminal  voltage  at  open  circuit,  and 
normalspeed,  as  function  of  the  ampere  turns  per  pole  field 
excitation. 

Such  curves  are  of  the  shape  shown  in  Fig.  90  as  A. 
Owing  to  the  remanent  magnetism  or  hysteresis  of  the  iron 
part  of  the  magnetic  circuit,  the  saturation  curve  taken  with 
decreasing  field  excitation  usually  does  not  coincide  with 
that  taken  with  increasing  field  excitation,  but  is  higher,  and 
by  gradually  first  increasing  the  field  excitation  from  zero 


COMMUTATING  MACHINES. 


189 


to  maximum  and  then  decreasing  again,  the  looped  curve  in 
Fig.  91  is  derived,  giving  as  average  saturation  curve  the 
curve  shown  in  Fig.  90  as  A,  and  as  central  curve  in 
Fig.  91. 


Fig.  90. 

Direct  current  generators  are  usually  operated  at  a  point 
of  the  saturation  curve  above  the  bend,  that  is,  at  a  point 
where  the  terminal  voltage  increases  considerably  less  than 
proportionally  to  the  field  excitation.  This  is  necessary  in 
self-exciting  direct  current  generators  to  secure  stability. 

The  ratio, 

increase  of  field  excitation  ^  corresponding  increase  of  voltage . 

total  field  excitation  total  voltage 

that  is,  /#F0  t  de 


is  called  saturation  coefficient  s,  and  is  plotted  in  Fig.  90 
with  the  voltage  as  ordinates,  and  the  saturation  coefficient 
s  as  abscissae. 

Of  considerable  importance  also  are  curvet  giving 
the  terminal  voltage  as  function  of  the  field  excitation  at 
load.  Such  curves  are  called  load  saturation  curves,  and  can 
be  constant  current  load  saturation  curve,  that  is,  terminal 


190 


ELECTRICAL   ENGINEERING. 


voltage  as  function  of  field  ampere  turns  at  constant  full-load 
current  flowing  through  the  armature,  and  constant  resist- 
ance load  saturation  curve,  that  is,  terminal  voltage  as 
function  of  field  ampere  turns  if  the  machine  circuit  is  closed 


TUR 


CUR 


ES. 


Fig.  91. 

through    a    constant   resistance   giving  full-load  current  at 
full-load  terminal  voltage. 

A  constant  current  load  saturation  curve  is  shown  as  B, 
and  a  constant  resistance  load  saturation  curve  as  C  in 
Fig.  90. 

IK.   Compounding. 

In  the  direct  current  generator  the  field  excitation  re- 
quired to  maintain  constant  terminal  voltage  has  to  be 
increased  with  the  load.  A  curve,  giving  the  field  excitation 


COMMUTATIA'G  MACHINES.  191 

in  ampere  turns  per  pole,  as  function  cf  the  load  in  amperes, 
at  constant  terminal  voltage,  is  called  the  compounding  cmve 
of  the  machine. 

The  increase  of  field  excitation  required   with    load    is 
due  to  : 

1.  The  internal  resistance  of  the  machine,  which  con- 
sumes E.M.F.  proportional  to  the  current,  so  that   the  in- 
duced  E.M.F.,  and  thus  the    field    M.M.F.    corresponding 
thereto,  has  to  be  greater   under   load.       If  /  =  resistance 

Z/'* 

drop  in  the  machine  as  fraction  of  terminal  voltage,  or  =  — ' 

e 

the  induced  E.M.F.  at  load  has  to  be  e  (1  4-  /),  and  if  £F0  = 
no-load  field  excitation,  and  s  =  saturation  coefficient,  the 
field  excitation  required  to  produce  the  E.M.F.  *•(!  +  /)  is 
(F0(l  +  st),  thus  an  additional  excitation  of  st$Q  is  required 
at  load,  due  to  the  armature  resistance. 

2.  The  demagnetizing  effect  of  the  ampere  turns  arma- 
ture reaction  of  the  angle  of  shift  of  brushes  requires  an  in- 
crease cf  field  excitation  by  /if.     (Section  VII.) 

3.  The  distorting  effect  of  armature  reaction  does  not 
change  the  total  M.M.F.  producing  the  magnetic  flux.       If, 
however,  magnetic  saturation  is  reached  or  approached  in  a 
part  of  the  magnetic  circuit  adjoining  the  air  gap,  the  in- 
crease of  magnetic  density  at  the  strengthened  pole  corner- 
is  less  than  the  decrease  at  the  weakened  pole  corner,  and 
thus  the  total  magnetic  flux  with  the    same  total  M.M.F. 
reduced,  and  to  produce  the  same  total  magnetic  flux  an  in- 
creased total  M.M.F.,  that  is,  increase  of  field  excitation,  is 
required.     This  increase  depends  upon  the  saturation  of  the 
magnetic  circuit  adjacent  to  the  armature  conductors. 

4.  The  magnetic  stray  field  of  the  machine,  that  is,  that 
part  of  the  magnetic  flux  which  passes  from  field  pole  to 
field  pole  without  entering  the  armature,  usually  increases 
with  the  load.     This  stray  field  is  proportional  to  the  dif- 
ference of  magnetic  potential  between  field  poles.     That  is, 
at   no-load  it  is  proportional  to  the  ampere  turns  M.M.F. 


192  ELECTRICAL   ENGINEERING. 

consumed  in  air  gap,  armature  teeth,  and  armature  core. 
Under  load,  with  the  same  induced  E.M.F.,  that  is,  the  same 
magnetic  flux  passing  through  the  armature  core,  the  dif- 
ference of  magnetic  potential  between  adjacent  field  poles  is 
increased  by  the  counter  M.M.F.  of  the  armature  and  the 
saturation.  Since  this  magnetic  stray  flux  passes  through 
field  poles  and  yoke,  the  magnetic  density  therein  is  in- 
creased and  the  field  excitation  correspondingly,  especially 
if  the  magnetic  density  in  field  poles  and  yoke  is  near 
saturation.  This  increase  of  field  strength  required  by  the 
increase  of  density  in  the  external  magnetic  circuit,  due  to 
the  increase  of  magnetic  stray  field,  depends  upon  the  shape 
of  the  magnetic  circuit,  the  armature  reaction,  and  the 
saturation  of  the  external  magnetic  circuit. 

Curves  giving,  with  the  amperes  output  as  abscissae,  the 
ampere  turns  per  pole  field  excitation  required  to  increase 
the  voltage  proportionally  to  the  current,  are  called  over- 
compounding  curves.  In  the  increase  of  field  excitation 
required  for  over-compounding,  the  effects  of  magnetic 
saturation  are  still  more  marked. 

X.   Characteristic  Curves. 

The  field  characteristic  or  regulation  cuive,  that  is,, 
curve  giving  the  terminal  voltage  as  function  of  the  current 
output  at  constant  field  excitation,  is  of  less  importance  in 
commutating  machines  than  in  synchronous  machines,  since 
commutating  machines  are  usually  not  operated  with  sepa- 
rate and  constant  excitation,  and  the  use  of  the  series  field 
affords  a  convenient  means  of  changing  the  field  excitation 
proportionally  to  the  load.  The  curve  giving  the  terminal 
voltage  as  function  of  current  output,  in  a  compound- wound 
machine,  at  constant  resistance  in  the  shunt  field,  and  con- 
stant adjustment  of  the  series  field,  is,  however,  of  impor- 
tance as  regulation  curve  of  the  direct  current  generator. 
This  curve  would  be  a  straight  line  except  for  the  effect  of 
saturation,  etc.,  as  discussed  above. 


COMMUTATING   MACHINES.  193 

XI.   Efficiency  and  Losses. 

The  losses  in  a  commutating  machine  which  have  to  be 
considered  when  deriving  the  efficiency  by  adding  the  indi- 
vidual losses  are  : 

1.  Loss  in  the  resistance  of  the  armature,  the  commu- 
tator leads,  brush  contacts  and  brushes,  in  the  shunt  field 
and  the  series  field  with  their  rheostats. 

2.  Hysteresis  and  eddy  currents  in  the  iron  at  a  voltage 
equal  to  the  terminal  voltage,  plus  resistance  drop  in  a  gene- 
rator, or  minus  resistance  drop  in  a  motor. 

3.  Eddy  currents  in  the  armature  conductors  when  large 
and  not  protected. 

4.  Friction  of  bearings,  of  brushes  on  the  commutator, 
and  windage. 

5.  Load  losses,  due  to  the  increase  of  hysteresis  and  of 
eddy    currents    under    load,  caused  by  the  change  of   the 
magnetic  distribution,  as  local  increase  of  magnetic  density 
and  of  stray  field. 

The  friction  of  the  brushes  and  the  loss  in  the  contact 
resistance  of  the  brushes,  are  frequently  quite  considerable, 
especially  with  low-voltage  machines. 

Constant  or  approximately  constant  losses  are:  Friction 
of  bearings  and  of  commutator  brushes  and  windage,  hyste- 
resis and  eddy  current  losses,  shunt  field  excitation.  Losses 
increasing  with  the  load,  and  proportional  or  approximately 
proportional  to  the  square  of  the  current :  those  due  to 
armature  resistance,  resistance  of  series  field,  resistance  of 
brush  contact,  and  the  so-called  "  load-losses,"  which,  how- 
ever, are  usually  small  in  commutating  machines. 

XII.   Commutation. 

The  most  important  problem  connected  with  commu- 
tating machines  is  that  of  commutation. 

Fig.    92   represents    diagrammatically    a    commutating 


ELECTRICAL   ENGINEERING. 

machine.  The  E.M.F.  induced  in  an  armature  coil  A  is 
zero  with  this  coil  at  or  near  the  position  of  the  commutator 
brush  BY  It  rises  and  reaches  a  maximum  about  midway 
between  two  adjacent  sets  of  brushes,  Bv  and  />2,  at  C,  and 
then  decreases  again,  reaching  zero  at  or  about  By  and  then 
repeats  the  same  change  in  opposite  direction.  The  current 


Fig.  92. 

in  armature  coil  A,  however,  is  constant  during  the  motion 
of  the  coil  from  Bl  to  By  While  the  coil  A  passes  the 
brush  By  however,  the  current  in  the  coil  A  reverses,  and 
then  remains  constant  again  in  opposite  direction  during  the 
motion  from  Bz  to  By  Thus,  while  the  armature  coils  of  a 
commutating  machine  are  the  seat  of  a  system  of  polyphase 
E.M.F.'s  having  as  many  phases  as  coils,  the  current  flow- 
ing in  these  coils  is  constant,  reversing  successively. 

The  reversal  of  current  in  coil  A  takes  place  while  the 


COMMUTAT1NG   MACHINES.  195 

gap  G  between  the  two  adjacent  commutator  segments 
between  which  the  coil  A  is  connected,  passes  the  brush  B^ 
Thus,  if  w  =  width  of  brushes,  s  =  peripheral  speed  of 
commutator  per  second  in  the  same  measure  in  which  w  is 
given,  as  in  inches  per  second  if  w  is  given  in  inches,  tQ  =  w/s 
is  the  time  during  which  the  current  in  A  reverses.  Thus, 
considering  the  reversal  as  a  single  alternation,  /o  is  a  half 

period,  and  thus  JVo  =—  — =^ —  is  the  frequency  of  comma- 
2/€      LW 

tation  ;  hence,  if  L  —  inductance  or  co-efficient  of  self-induc- 
tion of  the  armature  coil  A,  the  E.M.F.  induced  in  the 
armature  coil  during  commutation  is  eQ=  2WVoZz'o,  where 
zo=  current  reversed,  and  the  energy  which  has  to  be  dissi- 
pated during  commutation  is  i\L. 

The  frequency  of  commutation  is  very  much  higher  than 
the  frequency  of  synchronous  machines,  and  averages  from 
300  to  1000  cycles  per  second,  or  more. 

In  reality,  however,  the  changes  of  current  during  com- 
mutation are  not  sinusoidal,  but  a  complex  exponential 
function,  and  the  resistance  of  the  commutated  circuit  enters 
the  problem  as  an  important  factor.  In  the  moment  when 
the  gap  G  of  the  armature  coil  A  reaches  the  brush  By  the 
coil  A  is  short  circuited  by  the  brush,  and  the  current  ?0  in 
the  coil  begins  to  die  out,  or  rather  to  change  at  a  rate  de- 
pending upon  the  internal  resistance  and  the  inductance  of 
the  coil  Aj  and  the  E.M.F.  induced  in  the  coil  by  the  field 
magnetic  flux.  The  higher  the  internal  resistance,  the  faster 
is  the  change  of  current,  and  the  higher  the  inductance,  the 
slower  the  current  changes.  Thus  two  cases  have  to  be 
distinguished. 

1.  No  magnetic  flux  enters  the  armature  at  the  position 
of  the  brushes,  that  is,  no  E.M.F.  is  induced  in  the  armature 
coil  under  commutation,  except  that  of  its  own  self-induction. 
In  this  case  the  commutation  is  entirely  determined  by  the 
inductance  and  resistance  of  the  armature  coil  A>  and  is 
called  "Resistance  Commutation." 


196  ELECTRICAL   ENGINEERING. 

2.  The  brushes  are  shifted  so  that  commutation  takes 
place  in  an  active  magnetic  field.  That  is,  in  the  armature 
coil  during  commutation,  an  E.M.F.  is  induced  by  its  rota- 
tion through  the  magnetic  field  of  the  machine.  In  this  case 
the  commutation  depends  upon  the  inductance  and  the  re- 
sistance of  the  armature  coil,  and  the  E.M.F.  induced  therein 
by  the  main  magnetic  field,  and  is  called  "  Voltage  Com- 
mutation." 

In  either  case,  the  resistance  of  the  brushes  and  their 
contact  may  either  be  negligible,  as  usually  the  case  with 
copper  brushes,  or  it  may  be  of  the  same  or  a  higher  magni- 
tude than  the  internal  resistance  of  the  armature  coil  A. 
The  latter  is  usually  the  case  with  carbon  or  graphite 
brushes. 

In  the  former  case,  the  resistance  of  the  short  circuit  of 
armature  coil  A  under  commutation  is  approximately  con- 
stant ;  in  the  latter  case  it  varies  from  infinity  in  the  moment 
of  beginning  commutation  down  to  minimum,  and  then  up 
again  to  infinity  at  the  end  of  commutation. 

a.}    Negligible  resistance  of  brush  and  brush  contact. 

This  is  more  or  less  approximately  the  case  with  copper 
brushes 

Let  /0  =  current, 

L  =  inductance, 
r  —  resistance  of  armature  coil, 

w 
•  /0  =  —  =  time  of  commutation, 

and  —  e  =  E.M.F.  induced  in  the  armature  coil  by  its  rota- 
tion through  the  magnetic  field,  or  commutating  E.M.F. 

Denoting  the  current  in  the  coil  A  at  time  t  after  begin- 
ning of  commutation,  by  z,  the  E.M.F.  of  self-induction  is, 

di 


Thus  the  total  E.M.F.  acting  in  coil  A, 


COMMUTATING  MACHINES.  197 

and  the  current, 

.  _  —  e  -f  ev  _         e      L  di 
r  ~  ~r~  ~r  ~di 

Transposed,  this  expression  becomes, 
_  /v#_      di 

*-r  +  i 
the  integral  of  which  is, 

rt  le        \ 

—  -j-  =  loge   -  + 1)  —  log,  c, 
**  V         / 

where  \ogec  =  integration  constant. 
Since  at 

t=(\   i=iw 
we  have, 

(e        \ 
r  +  *°)  ' 
therefore, 


hence  : 


and,  at  the  end  of  commutation,  or,  t  =  /0, 


for  perfect  commutation,  it  must  be, 


that  is,  the  current  at  the  end  of  commutation  must  have 
reversed  and  reached  its  full  value  in  opposite  direction. 

Substituting  in  this  last  equation  the  value  of  ^  from  the 
preceding  equation,  and  transforming,  we  have, 


198  ELECTRICAL    ENGINEERING. 

taking  the  logarithms  of  both  terms, 

e 


or  solving  the  exponential  equation  for  e,  we  obtain, 


It  is  evident  that  the  inequation  c  >  i^r  must  be  true,  other- 
wise perfect  commutation  is  not  possible. 

If 

•,=  0, 
we  have, 


That  is,  the  current  never  reverses,  but  merely  dies  out 
more  or  less,  and  in  the  moment  where  the  gap  G  of  the 
armature  coil  leaves  the  brush  f>,  the  current  therein  has  to 
rise  suddenly  to  full  intensity  in  opposite  direction.  This 
being  impossible,  due  to  the  inductance  of  the  coil,  the  cur- 
rent flows  as  arc  from  the  brush  across  the  commutator 
surface  for  a  length  of  time  depending  upon  the  inductance 
of  the  armature  coil, 

That  is,  with  low  resistance  brushes,  resistance  commuta- 
tion is  not  possible  except  with  machines  of  extremely  low 
armature  inductance,  that  is,  armature  inductance  so  low 

/  2Z, 
that  the  magnetic   energy  -^—  —  ,  which  appears  as  spark  in 

this  case,  is  harmless. 

Voltage  commutation  is  feasible  with  low  resistance 
brushes,  but  requires  a  commutating  E.M.F.  e  proportional 
to  current  z'0,  that  is,  requires  shifting  of  brushes  propor- 
tionally to  the  load. 

In  the  preceding,  the  E.M.F.,  e,  has  been  assumed  con- 
stant during  the  commutation.  In  reality  it  varies  some- 


COMMUTA  TING  MA  C HIKES. 


199 


what,  increasing  with  the  approach  of  the  commutated  coil 
to  a  denser  field.  It  is  not  possible  to  consider  this  varia- 
tion in  general,  and  c  is 
thus  to  be  considered 
the  average  value  dur- 
ing commutation. 

b.)  High  resistance 
brush  contact. 

Fig.  93  represents 
a  brush  B  commutating 
armature  coil  A. 

Let  r(}=  contact  re- 
sistance of  brush,  that 
is,  resistance  from  brush  to  commutator  surface  over  the 
total  bearing  surface  of  the  brushes.  The  resistance  of  the 
commutated  circuit  is  thus  internal  resistance  of  the  arma- 
ture coil  r>  plus  the  resistance  from  C  to  B,  plus  the  resist- 
ance from  B  to  D. 

Thus,  if  /0=  time  of  commutation,  at  the  time  /  after 
the  beginning  of  the  commutation,  the  resistance  from  C  to 

B  is  -"  °  and  from  B  to  D  is — ^L ,  thus,  the  total  resistance 
of  commutated  coil  is, 


Fig.  93. 


If  i(=  current  in  coil  A  before  commutation,  the  total 
current  entering  the  armature  from  brush  B  is  2  z'0.  Thus, 
if  i  =  current  in  commutated  coil,  the  current  ?0  4-  i  flows 
from  B  to  D,  the  current  z()  —  i  from  B  to  C. 

Hence,  the  difference  of  potential  from  D  to  C  is, 

^  +  0-^(4-0.  . 


The  E.M.F.  acting  in  coil  A  is, 


Ldi 

~w 


200  ELECTRICAL   ENGINEERING. 

and  herefrom  the  difference  of  potential  from  D  to  C, 

rdi  _  • 
-e-L--  ?r; 

hence, 

-e-L-t-ir=  -—  (/0  +  i)  -  -y  (/0  -  i). 

Or,  transposed, 


The  further  solution  of  this  general  problem  becomes 
difficult,  but  even  without  integrating  this  differential  equa- 
tion, a  number  of  important  conclusions  can  be  derived. 

Obviously  the  commutation  is  correct  and  thus  sparkless, 
if  the  current  entering  over  the  brush  shifts  from  segment 
to  segment  in  direct  proportion  to  the  motion  of  the  gap 
between  adjacent  segments  across  the  brush,  that  is,  if  the 
current  density  is  uniform  all  over  the  contact  surface  of 
the  brush.  This  means,  that  the  current  i  in  the  short- 
circuited  coil  varies  from  +  z'0  to  —  ?0  as  a  linear  function  of 
the  time.  In  this  case  it  can  be  represented  by, 


thus, 

dr  =    _2^ 
dt~     "4; 

Substituting  this  value  in  the  general  differential  equa 
tion,  gives,  after  some  transformation, 


or, 


which  gives  at  the  beginning  of  commutation,  t  —  0, 


COMMUTATING  MACHINES.  201 


'i  =  'o,    , 
<o 


at  the  end  of  commutation,  t  =  /0, 

.  /2Z 


That  is  : 

Even  with  high  resistance  brushes,  for  perfect  commu- 
tation, voltage  commutation  is  necessary,  and  the  E.M.F. 
e  impressed  upon  the  commutated  coil  must  increase  during 
commutation  from  el  to  ry  by  the  above  equation.  This 
E.M.F.  is  proportional  to  the  current  iv  but  is  independent 
of  -the  brush  resistance,  r0. 

RESISTANCE  COMMUTATION. 

Herefrom  it  follows,  that  resistance  commutation  cannot 
be  perfect,  but  that  at  the  contact  with  the  segment  that 
leaves  the  brush,  the  current  density  must  be  higher  than 
the  average.  Let  a  =  ratio,  of  actual  current  density  at  the 
moment  of  leaving  the  brush,  to  average  current  density  of 
brush  contact,  and  considering  only  the  end  of  commutation, 
as  the  most  important  moment,  we  have, 


*  —  *o  -  ~.  -  » 
'o 

since  for 

/=/„-/&. 
this  gives 

i  =  -  >0-h2a-/0, 
ro 

while  uniform  current  density  would  require, 

/  =  —  4  +  2-/0. 
'o 

The  general  differential  equation  of  resistance  commuta- 
tion, e  —  0,  is, 


202  ELECTRICAL   ENGINEERING. 

Substituting  in  this  equation,  the  value  of  i  from  the  fore- 
going equation,  expanding  and  cancelling  /0  —  /,  we  obtain, 


2r(/02(a-l)  +  r//0(2a-l)  -  2  ar/2  -  2  aLt  =  0. 

hence, 

/Q  (2  r0/0  +  r/) 


and  for 


That  is,  a  is  always  >  1. 

The  smaller  Z,  and  the  larger  r0,  the  smaller  is  a,  that  is, 
the  better  is-  the  commutation. 

Sparkless  commutation  is  impossible  for  very  large  values 
of  a,  that  is,  when  L  approaches  rQtQ,  or  when  r0  is  not  much 

larger  than  —  • 
'o 

XIII.   Types  of  Commutating  Machines. 

By  the  excitation,  commutating  machines  are  subdivided 
into  magneto,  separately  excited,  shunt,  series,  and  compound 
machines.  Magneto  machines  and  separately  excited  machines 
are  very  similar  in  their  characteristics.  In  either,  the  field 
excitation  is  of  constant,  or  approximately  constant,  im- 
pressed M.M.F.  Magneto  machines,  however,  are  little  used, 
and  only  for  very  small  sizes. 

By  the  direction  of  energy  transformation,  commutating 
machines  are  subdivided  into  generators  and  motors. 

Of  foremost  importance  in  discussing  the  different  types 
of  machines,  is  the  saturation  curve  or  magnetic  character- 
istic, that  is,  a  curve  relating  terminal  voltage  at  constant 
speed,  to  ampere  turns  per  pole  field  excitation,  at  open 
circuit.  Such  a  curve  is  shown  as  A  in  Figs.  94  and  95. 
It  has  the  same  general  shape  as  the  magnetic  induction 
curve,  except  that  the  knee  or  bend  is  less  sharp,  due  to 


COMMUTATING   MACHINES. 


203 


the   different  parts   of  the  magnetic  circuit  reaching   satu- 
ration successively. 

Thus,  to  induce  voltage  ac,  the  field  excitation  oc  is  re- 
quired. Subtracting  from  ac  in  a  generator,  Fig.  94,  or 
adding  in  a  motor,  Fig.  95,  the  value  ab  =  z>,  the  voltage 
consumed  by  the  resistance  of  armature,  commutator,  etc., 
gives  the  terminal  voltage  be  at  current  iy  and  adding  to  oc 


Fig.  94. 

the  value  ce  =  bd  =  iq  =  armature  reaction,  cr  rather  field 
excitation  required  to  overcome  the  armature  reaction,  gives 
the  field  excitation  oe  required  to  produce  the  terminal 
voltage  de  at  current  i.  The  armature  reaction,  ig,  corre- 
sponding to  current  i,  is  calculated  as  discussed  before, 
and  q  may  be  called  the  "Coefficient  of  Armature  Re- 
action." 

Such  a  curve,  D,  shown  in  Fig.  94  for  a  generator,  and 
in  Fig.  95  for  a  motor,  and  giving  the  terminal  voltage  de 
at  current  z,  corresponding  to  the  field  excitation  oe,  is  called 
a  Load  Saturation  Curve.  Its  points  are  respectively  dis- 
tant from  the  corresponding  points  of  the  no-load  satura- 


204 


ELECTRICAL   ENGINEERING. 


tion  curve  A    a  constant   distance    equal  to    ad,  measured 
parallel  thereto. 

Curves  D  are  plotted  under  the  assumption  that  the 
armature  reaction  is  constant.  Frequently,  however,  at 
lower  voltage  the  armature  reaction,  or  rather  the  increase 
of  excitation  required  to  overcome  the  armature  reaction,  2.7, 


Fig.  95. 

increases,  since  with  voltage  commutation  at  lower  voltage, 
and  thus  weaker  field  strength,  the  brushes  have  to  be 
shifted  more  to  secure  sparkless  commutation,  and  thus  the 
demagnetizing  effect  of  the  angle  of  lead  increases.  At 
higher  voltage  iq  usually  increases  also,  due  to  increase 
of  magnetic  saturation  under  load,  caused  by  the  increased 
stray  field.  Thus,  the  load  saturation  curve  of  the  con- 
tinuous current  generator  more  or  less  deviates  from  the 
theoretical  shape,  D  towards  a  shape  shown  as  G. 


COMMUTATJNG   MACHINES. 


205 


A.    GENERATORS. 
Separately  excited  and  Magneto  Generator. 

In  a  separately  excited  or  magneto  machine,  that  is,  a 
machine  with  constant  field  excitation,  CF0,  a  demagnetiza- 
tion curve  can  be  plotted  from  the  magnetization  or  satura- 
tion curved  in  Fig.  94.  At  current  t,  the  resultant  M.M. F, 
of  the  machine  is  fr^—iq,  and  the  induced  voltage  corre- 
sponds thereto  by  the  saturation  curve  A  in  Fig.  94.  Thus, 
in  Fig.  96  a  demagnetization  curve  A  is  plotted  with  the 


c 

EPA 

UT 

ELY 

EXC 

ITED  OR   M 

GN 

-.TO 

GE 

MER 

TOF 

?. 

r 

EM 

\GN 

rnzATic 

)N   C 

URVE  AND 

LOAD   C 

HAR 

kCTERIS 

TIC. 

CONST/ 

NT 

SHI 

"T   C 

F  BjRUS 

HES 

. 

^ 

—  —  -^ 

-~— 

—  -~- 

"-^-^^^ 

a 

^ 

\ 

~^-~. 

•^ 

*V 

"\. 

^ 

s    f\ 

RN 

N 

N 

\ 

\ 

s 

CO 

\ 

\ 

o 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

—AM 

PERE 

—  + 

\ 

\ 

0 

b 

| 

\ 

\ 

io 

Fig.  96. 

current  ob  =  i  as  abscissae,  and  the  induced  E.M.F.  ab  as 
ordinates,  under  the  assumption  of  constant  coefficient  of 
armature  reaction  q ;  that  is,  corresponding  to  curve  D  in 
Fig.  94.  This  curve  becomes  zero  at  the  current  z'0  which 
makes  i^q—  JF0.  Subtracting  from  curved  in  Fig.  32  the 
drop  of  voltage  in  the  armature  and  commutator  resistance, 
ac  =  ir,  gives  the  external  characteristic  B  of  the  machine  as 
generator,  or  the  curve  relating  the  terminal  voltage  to  the 
current. 


206 


ELEC  TRIG  A  L   ENG1NEEK1A  TG. 


In  Fig.  97  the  same  curves  are  shown  under  the  assump- 
tion that  the  armature  reaction  varies  with  the  voltage  in 
the  way  as  represented  by  curve  G  in  Fig.  94. 

In  a  separately  excited  or  magneto  motor  at  constant 
speed,  the  external  characteristic  would  lay  as  much  above 
the  demagnetization  curve  A  as  it  lays  below  in  a  generator 


EPA 

RAT 

ELY 

EXCITE 

D    0 

R   M 

\GN 

ITO 

GE^ 

ER 

TO 

3Efv 

GN 

ETIZ 

ATI 

DN   CUR 

VE* 

ND 

LOA 

DCF 

HAR/e 

CT 

RIS 

TIC. 

• 

VA 

RIABLE 

jHlFT   O 

F    BF 

USHES. 

fc- 

==: 

1  —  -—  . 

—  -  — 

---- 

—  ^^ 

^U 

S^ 

\ 

A 

N. 

B 

X 

s, 

V 

\ 

\ 

>j 

\ 

\ 

\ 

A 

\ 

\ 

- 

5 

\\ 

3 

V 

\ 

\ 

\ 

—  i  — 

AMPER 

* 

:s 

\\ 

\ 

Fig.  97. 

in  Fig.  96,  and  at  constant  voltage  the  speed  would  vary 
inversely  proportional  hereto. 

Shunt  Generator. 

The  external  or  load  characteristic  of  the  shunt  genera- 
tor is  plotted  in  Fig.  98  with  the  current  as  abscissae  and 
the  terminal  voltage  as  ordinates,  as  A  for  constant  co- 
efficient of  armature  reaction,  and  as  B  for  a  coefficient  of 
armature  reaction  varying  with  the  voltage  in  the  way  as 
shown  in  G,  Fig.  94,  The  construction  of  these  curves  is 
as  follows : 

In  Fig.  94,  og  is  the  straight  line  giving  the  field  excita- 


COMMUTATING   MACHINES. 


207 


tion  oh  as  function  of  the  terminal  voltage  Jig  (the  former 
obviously  being  proportional  to  the  latter  in  the  shunt 
machine).  The  open  circuit  or  no-load  voltage  of  the 
machine  is  then  kq. 

Drawing  gl  parallel  to  da  (assuming  constant  coefficient 
of  armature  reaction,  or  parallel  to  hypothenuse  of  the  tri- 
angle iqy  ir  at  voltage  og>  when  assuming  variable  armature 


Fig.  98. 

reaction),  then  the  current  which  gives  voltage  gh  is  pro- 
portional to  gl,  that  is,  i  :  full-load  current :  =  g I :  da. 

As  seen  from  Fig.  98,  a  maximum  value  of  current 
exists  which  is  less  if  the  brushes  are  shifted  than  at  con- 
stant position  of  brushes. 

From  the  load  characteristic  of  the  shunt  generator,  the 
resistance  characteristic  is  plotted  in  Fig.  99 ;  that  is,  the 
dependence  of  the  terminal  voltage  upon  the  external  resist  - 

terminal  voltage 

ance  R  =  -  • —    Curve  A  in  Fig.  99  corresponds 

current. 

to  constant,  curve  B  to  varying  armature  reaction.  As  seen, 
at  a  certain  definite  resistance,  the  voltage  becomes  zero, 


208 


ELECTRICAL   ENGINEERING. 


and  for  lower  resistance  the  machine  cannot  generate  but 
loses  its  excitation. 

The  variation  of  the  terminal  voltage  of  the  shunt  gen- 
erator with  the  speed  at  constant  field  resistance  is  shown 
in  Fig.  100  at  no  load  as  A,  and  at  constant  current  it  as  B. 
These  curves  are  derived  from  the  preceding  ones.  They 


Fig.  99. 

show  that  below  a  certain  speed,  which  is  much  higher  at 
load  than  at  no  load,  the  machine  cannot  generate.  The 
lower  part  of  curve  B  is  unstable  and  cannot  be  realized. 

Series  Generator. 

In  the  series  generator  the  field  excitation  is  proportional 
to  the  current  z,  and  the  saturation  curve  A  in  Fig.  101  can 
thus  be  plotted  with  the  current  z  as  abscissae.  Subtracting 
ab  =  z>,  the  resistance  drop,  from  the  voltage,  and  adding 
bd  =  iq,  the  armature  reaction,  gives  a  load  saturation  curve 
or  external  characteristic  B  of  the  series  generator.  The 
terminal  voltage  is  zero  at  no  load  or  open  circuit,  increases 


COMMUTATING  MACHINES. 


209 


with  the  load,  reaches  a  maximum  value  at  a  certain  current, 
and  then  decreases  again  and  reaches  zero  at  a  certain 
maximum  current,  the  current  of  short  circuit. 


SHUNT   <p 
SPEED   CHARACTERS 


1C. 


ON.STANT 


FIELD   RESISTANCE. 


Fig.  100. 

Curve  B  is  plotted  with  constant  coefficient  of  armature 
reaction  q.  Assuming  the  brushes  to  be  shifted  with  the 
load  and  proportionally  to  the  load,  gives  curves  C,  D,  and  Ey 


210 


ELECTRICAL   ENGINEERING. 


which  are  higher  at  light-  load,  but  fall  off  faster  at  high  load. 
A  still  further  shift  of  brushes  near  the  maximum  current 
value  even  overturns  the  curve  as  shown  in  F.  Curves  E 
and  F  correspond  to  a  very  great  shift  of  brushes,  and  an 
armature  demagnetizing  effect  of  the  same  magnitude  as 
the  field  excitation,  as  realized  in  arc-light  machines,  in 
which  the  last  part  of  the  curve  is  used  to  secure  inherent 
regulation  for  constant  current. 


Fig.   101. 

The  resistance  characteristic,  that  is,  the  dependence  of 
the  current  and  of  the  terminal  voltage  of  the  series  gener- 
ator upon  the  external  resistance,  are  constructed  from  Fig. 
102  and  plotted  in  Fig.  102. 

BI  and  B2,  in  Fig.  102,  are  terminal  volts  and  amperes 
corresponding  to  curve  B  in  Fig.  101,  Ev  Ey  and  F2  volts 
and  amperes  corresponding  to  curves  E  and  Fin  Fig.  101. 

Above  a  certain  external  resistance  the  series  generator 
loses  its  excitation,  while  the  shunt  generator  loses  its  exci- 
tation below  a  certain  external  resistance. 


COMMUTA  TING   MA  CHINES. 


211 


Compound  Generator. 

The  saturation  curve  or  magnetic  characteristic  A,  and 
the  load  saturation  curves  D  and  G  of  the  compound  gen- 
erator, are  shown  in  Fig.  103  with  the  ampere  turns  of  the 


\ 


NERATCR 


\\ 


RESISTANCE 


HARACTERISTIC 


RESISTANCE 


\ 


Fig.  102. 


Fig.  103. 


212 


ELECTRICAL   ENGINEERING 


shunt  field  as  abscissae.  A  is  the  same  curve  as  in  Fig.  94, 
while  D  and  G  in  Fig.  103  are  the  corresponding  curves  of 
Fig.  94  shifted  to  the  left  by  the  distance  iqv  the  M.M.F. 
of  ampere  turns  of  the  series  field. 

At  constant  position  of  brushes  the  compound  generator, 
when  adjusted  for  the  same  voltage  at  no  load  and  at  full 
load,  under-compounds  at  higher  and  over-compounds  at 
lower  voltage,  and  even  at  open  circuit  of  the  shunt  field, 
gives  still  a  voltage  op  as  series  generator.  When  shifting 
the  brushes  under  load,  at  lower  voltage  a  second  point  g  is 
reached  where  the  machine  compounds  correctly,  and  below 
this  point  the  machine  under-compounds  and  loses  its  exci- 
tation when  the  shunt  field  decreases  below  a  certain  value  ; 
that  is,  it  does  not  excite  itself  as  series  generator. 


B.     MOTORS. 
Shunt  Motor. 

Three  speed  characteristics  of  the  shunt  motor  at  con- 
stant impressed  E.M.F.  e  are  shown  in  Fig.  104  as  A,  P,  Qt 


TTTJNTTv  OTOR 
PEEP  C  JRVES. 

kNT     MPR    RfiF       V( 


CONSTANT    MPR 


S8E     VO  TAG 


Fig.  104. 


corresponding  to  the  points  d,  /,  q  of  the  motor  load  satu- 
ration curve,  Fig.  95.  Their  derivation  is  as  follows :  At 
constant  impressed  E.M.F.  e,  the  field  excitation  is  constant 


COMMUTATING  MACHINES. 


213 


=  FO,  and  at  current  i  the  induced  E.M.F.  must  be  e  —  ir. 
The  resultant  field  excitation  is  (F0  —  iq,  and  corresponding 
hereto  at  constant  speed  the  induced  E.M.F.  taken  from 
saturation  curve  A  in  Fig.  95  is  ev  Since  it  must  be  e  —  ir, 

£  —  IF 

the  speed  is  changed  in  the  proportion  -      — 

e\ 
At  a  certain  voltage  the  speed  is  very  nearly  constant, 

the  demagnetizing  effect  of  armature  reaction  counteracting 
the  effect  of  armature  resistance.  At  higher  voltage  the 
speed  falls,  at  lower  voltage  it  rises  with  increasing  current. 


/ 

/ 

. 

x1 

/ 

* 

/ 

/ 

' 

/ 

t 

/ 

SH 

UNI 

MO 

TOF 

3 

/ 

/ 

SF 

EED 

CU 

RVE 

\ 

/ 

* 

/API, 

BLE 

IMP 

ESS 

D    E 

M. 

t 

/ 

VOL 

rs 

•> 

Fig.  105. 

In  Fig.  105  is  shown  the  speed  characteristic  of  the  shunt 
motor  as  function  of  the  impressed  voltage  at  constant  out- 
put ;  that  is,  constant  product,  current  times  induced  E.M.F. 

If  i  =  current  and  P  =  constant  output,  the  induced  E.M.F. 

P 
must  be  approximately  el  =  —>  and  thus  the  terminal  voltage 

e  =  ^  +  ir.  Proportional  hereto  is  the  field  excitation  £F0. 
The  resultant  M.M.F.  of  the  field  is  thus,  &  =  ^F0  —  iq,  and 
•corresponding  thereto  from  curve  A  in  Fig.  96  the  E.M.F. 
^o  is  derived  which  would  be  induced  at  constant  speed  by 
M.M.F.  JF. 

Since,  however,  the  induced  E.M.F.  must  be  el  the  speed 

g 

is  changed  in  the  proportion  — • 


214 


ELECTRICAL   ENGINEERING. 


The  speed  rises  with  increasing,  and  falls  with  decreasing 
impressed  E.M.F.  With  still  further  decreasing  impressed 
E.M.F.,  the  speed  reaches  a  minimum  and  then  increases, 
again,  but  the  conditions  become  unstable. 

Series  Motor. 

The  speed  characteristic  of  the  series  motor  is  shown  in 
Fig.  106  at  constant  impressed  E.M.F.  e.  A  is  the  saturation 


IRIES  MOTOR 


SPEED  CURVE, 


Fig.  106. 

curve  of  the  series  machine,  with  the  current  as  abscissae 
and  at  constant  speed.     At  current  i,  the  induced  E.M.F. 


must  be  e  —  ir,  and  the  speed  is  thus 


times  that,  for 


which  curve  A  is  plotted  where  el  =  E.M.F.  taken  from 
saturation  curve  A.  This  speed  curve  corresponds  to  a  con- 
stant position  of  brushes  midway  between  the  field  poles,  as 
generally  used  in  railway  motors  and  other  series  motors.  If 
the  brushes  have  a  constant  shift  or  are  shifted  propor- 
tionally to  the  load,  instead  of  the  saturation  curve  A  in  Fig. 
106,  a  curve  is  to  be  used  corresponding  to  the  position  of 


COM  MUTATING   MACHINES.  215 

brushes,  that  is,  derived  by  adding  to  the  abscissae  of  A  the 
values  iq,  the  demagnetizing  effect  of  armature  reaction. 

The  torque  of  the  series  motor  is  shown  also  in  Fig.  106, 
derived  as  proportional  to  A  x  z,  that  is,  current  x  magnetic 
flux. 

Compound  Motors. 

Compound  motors  can  be  built  with  cumulative  com- 
pounding and  with  differential  compounding. 

Cumulative  compounding  is  used  to  a  considerable  ex- 
tent as  in  elevator  motors,  etc.,  to  secure  economy  of  current 
in  starting  and  at  high  loads  at  the  sacrifice  of  speed  regula- 
tion. That  is,  a  compound  motor  with  cumulative  series 
field  stands  in  its  speed  and  torque  characteristic  intermedi- 
ate between  the  shunt  motor  and  series  motor. 

Differential  compounding  is  used  to  secure  constancy  of 
speed  with  varying  load,  but  to  a  small  extent  only,  since  the 
speed  regulation  of  a  shunt  motor  can  be  made  sufficiently 
close  as  was  shown  in  the  preceding. 

Conclusion. 

The  preceding  discussion  of  commutating  machine  types 
can  obviously  be  only  very  general,  showing  the  main 
characteristics  of  the  curves,  while  the  individual  curves  can 
be  modified  to  a  considerable  extent  by  suitable  design  of  the 
different  parts  of  machine  when  required  to  derive  certain 
results,  as  for  instance  to  extend  the  constant  current  part 
of  the  series  generator,  or  to  derive  a  wide  range  of  voltage 
at  stability,  that  is,  beyond  the  bend  of  the  saturation  curve 
in  the  shunt  generator,  or  to  utilize  the  range  of  the  shunt 
generator  load  characteristic  at  the  maximum  current  point 
for  constant  current  regulation,  cr  to  secure  constancy  of 
speed  in  a  shunt  motor  at  varying  impressed  E.M.F.,  etc. 

The  use  of  the  commutating  machine  as  direct  cur- 
rent converter  has  been  omitted  from  the  preceding  dis- 


216  ELECTRICAL   ENGINEERING. 

cussion.  By  means  of  one  or  more  alternating  current 
compensators  or  auto-transformers,  connected  to  the  arma- 
ture by  collector  rings,  the  commutating  machine  can  be 
used  to  double  or  halve  the  voltage,  or  convert  from  one  side 
of  a  three-wire  system  to  the  other  side.  Since,  however, 
the  direct  current  converter  exhibits  many  features  simi- 
lar to  those  of  the  synchronous  converter,  as  regards  the 
absence  of  armature  reaction,  the  reduced  armature  heating, 
etc.,  it  will  be  discussed  as  an  appendix  to  the  synchronous 
converter. 


SYNCHRONOUS   CONVERTERS.  217 


C.  SYNCHRONOUS  CONVERTERS. 
I.  General. 

FOR  long  distance  transmission,  and  to  a  certain  extent 
also  for  distribution,  alternating  currents,  either  polyphase 
or  single-phase,  are  extensively  used.  For  many  applications, 
however,  as  especially  for  electrical  railroading  and  for 
electrolytic  work,  direct  currents  are  required,  and  are 
usually  preferred  also  for  low-tension  distribution  on  the 
Edison  three-wire  system.  Thus,  where  power  is  derived 
from  an  alternating  system,  transforming  devices  are  required 
to  convert  from  alternating  to  direct  current.  This  can  be 
done  either  by  a  direct  current  generator  driven  by  an  alter- 
nating synchronous  or  induction  motor,  or  by  a  single 
machine  consuming  alternating  and  producing  direct  current 
in  one  and  the  same  armature.  Such  a  machine  is  called  a 
converter,  and  combines,  to  a  certain  extent,  the  features  of 
a  direct  current  generator  and  an  alternating  synchronous 
motor,  differing,  however,  from  either  in  other  features. 

Since  in  the  converter  the  alternating  and  the  direct  cur- 
rent pass  through  the  same  armature  conductors,  their 
E.M.F.'s  stand  in  a  definite  relation  to  each  other,  which  is 
such  that  in  practically  all  cases  step-down  transformers  are 
necessary  to  generate  the  required  alternating  voltage. 

Comparing  thus  the  converters  with  the  combination  of 
synchronous  or  induction  motor  and  direct  current  generator, 
the  converter  requires  step-down  transformers,  —  the  syn- 
chronous motor,  if  the  alternating  line  voltage  is  considerably 
above  10,&)0  volts,  generally  requires  step-down  transformers 
also,  —  with  voltages  of  1000  to  10,000  volts,  however,  the 
synchronous  motor  can  frequently  be  wound  directly  for  the 


218  ELECTRICAL   ENGINEERING. 

line  voltage  and  stationary  transformers  saved.  Thus,  on 
the  one  side  we  have  two  machines  with  or  generally  without 
stationary  transformers,  on  the  other  side  a  single  machine 
with  transformers. 

Regarding  the  reliability  of  operation  and  first  cost, 
obviously  a  single  machine  is  preferable. 

Regarding  efficiency,  it  is  sufficient  to  compare  the  con- 
verter with  the  synchronous-motor-direct-current-generator 
set,  since  the  induction  motor  is  inherently  less  efficient  than 
the  synchronous  motor.  The  efficiency  of  the  stationary 
transformer  of  large  size  varies  from  97%  to  98.%,  with  an 
average  of  97.5%.  That  of  the  converter  or  of  the  synchro- 
nous motor  varies  between  91%  and  95%,  with  93%  as  ave- 
rage, and  that  of  the  direct  current  generator  between  90% 
and  94%,  with  92%  as  average.  Thus  the  converter  with  its 
step-down  transformers  will  give  an  average  efficiency  of 
90.7%,  a  direct  current  generator  driven  by  synchronous 
motor  with  step-down  transformers  an  efficiency  of  83.4%, 
without  step-down  transformers  an  efficiency  of  85.6%. 
Hence,  the  converter  is  more  efficient. 

Mechanically  the  converter  has  the  advantage  that  no 
transfer  of  mechanical  energy  takes  place,  since  the  torque 
consumed  by  the  generation  of  the  direct  current  and 
the  torque  produced  by  the  alternating  current  are  applied 
at  the  same  armature  conductors,  while  in  a  direct  current 
generator  driven  by  synchronous  motor  the  power  has  to  be 

transmitted  mechanically  through  the  shaft. 

-> 

II.   Ratio  of  E.M.F.'s  and  of  Currents. 

In  its  structure,  the  synchronous  converter  consists  of 
a  closed  circuit  armature,  revolving  in  a  direct  current- 
excited  field,  and  connected  to  a  segmental  commutator  as 
well  as  to  collector  rings.  Structurally,  it  thus  differs  frcm 
a  direct  current  machine  by  the  addition  of  the  collector 
rings,  from  certain  (very  little  used)  forms  of  synchronous 
.machines  by  the  addition  of  the  segmental  commutator. 


S  YNCHRONOUS   CONVER TERS. 


219 


In  consequence  hereof,  regarding  types  of  armature 
windings  and  of  field  windings,  etc.,  the  same  rule  applies  to 
the  converter  as  to  all  commutating  machines,  except  that 
in  the  converter  the  total  number  of  armature  coils  with  a 
series-wound  armature,  and  the  number  of  armature  coils 
per  pair  of  poles  with  a  multiple-wound  armature,  should  be 
divisible  by  the  number  of  phases. 

Regarding  the  wave-shape  of  the  alternating  induced 
E.M.F.,  similar  considerations  apply  as  for  a  synchronous 
machine  with  closed  circuit  armature ;  that  is,  the  induced 
E.M.F.  usually  approximates  a  sine  wave,  due  to  the  multi- 
tooth  distributed  winding. 

Thus,  in  the  following,  only  those  features  will  be  dis- 
cussed in  which  the  synchronous  converter  differs  from  the 
commutating  machines 
and  synchronous  machines 
treated  in  the  preceding 
chapter. 

Fig.  107  represents 
ctiagrammatically  the  com- 
mutator of  a  direct  cur- 
rent machine  with  the  ar- 
mature coils  A  connected 
to  adjacent  commutator  Flf- 107- 

bars.     The  brushes  are  BJ$V  and  the  field  poles 

If  now  two  oppositely  located  points  a^  of  the  commu- 
tator are  connected  with  two  collector  rings  D^D^  it  is 
obvious  that  the  E.M.F.  between  these  points  a^av  and  thus 
between  the  collector  rings  D^DV  will  be  a  maximum  in  the 
moment  where  the  points  a^2  coincide  with  the  brushes 
B^By  and  is  in  this  moment  equal  to  the  direct  current 
voltage  E  of  this  machine.  While  the  points  a^2  move 
away  from  this  position,  the  difference  of  potential  between 
#!  and  az  decreases  and  becomes  zero  in  the  moment  where 
a^2  coincide  with  the  direction  of  the  field  poles,  F^.  In 
this  moment  the  difference  in  potential  between  CL  and  a2 


220  ELECTRICAL   ENGINEERING. 

reverses  and  then  increases  again,  reaching  equality  with  E, 
but  in  opposite  direction,  when  al  and  #2  coincide  with  the 
brushes  B^  and  Br 

That  is,  between  the  collector  rings  Z\  and  D2  an  alter- 
nating voltage  is  produced,  whose  maximum  value  equals  the 
direct  current  electro-motive-force  E,  and  which  makes  a 
complete  period  for  every  revolution  of  the  machine  (in  a 
bipolar  converter,  or  p  periods  per  revolution  in  a  machine 
2/  poles). 

Hence,  this  alternating  E.M.F.  is, 

e  =  E  sin  2  ttNt. 
Where  N=-  frequency  of  rotation, 

£  =  E.M.F.  between  brushes  of  the  machine. 

Thus,  the  effective  value  of  the  alternating  E.M.F.  is, 


That  is,  a  direct  current  machine  produces  between  two 
collector  rings  connected  with  two  opposite  points  of  the 

commutator,  an  alternating  E.M.F.  of  —  x  the  direct  cur- 

V2 

rent  voltage,  at  a  frequency  equal  to  the  frequency  of 
rotation,  and  since  every  alternating  current  generator  is 
reversible,  such  a  direct  current  machine  with  two  collector 

rings,  when  supplied  with  an  alternating  E.M.F.  of  —  -x  the 

V2   ' 

direct  current  voltage  at  the  frequency  of  rotation,  will  run 
as  synchronous  motor,  or  if  at  the  same  time  generating 
direct  current,  as  synchronous  converter. 

Since,  neglecting  losses  and  phase  displacement,  the  out- 
put of  the  direct  current  side  must  be  equal  to  the  input  of 
the  alternating  current  side,  and  the  alternating  voltage  in 

the  single-phase  converter  is  —  —  X  E,  the  alternating  current 

V2 

must  be  =  V2  x  /,  where  /  =  diiect  current  output. 

If  now  the  commutator  is  connected  to  a  further  pair  of 


SYNCHRONOUS   CONVERTERS. 


221 


collector  rings,  D3  D4  (Fig.  108)  at  the  points  as  and  a4  midway 

between  al  and  <?2,  it  is  obvious  that  between  Dz  and  D4  an 

alternating    voltage    of    the    same    fre- 

quency and  intensity  will  be  produced 

as   between  Dl  and  Z?2,  but  in  quadra- 

ture   therewith,    since    at    the    moment 

where    az    and    a4    coincide    with    the 

brushes    Bl    Bz   and    thus    receive    the 

maximum  difference  of  potential,  a^  and 

<z2  are  at  zero  points  of  potential. 

Thus    connecting    four    equidistant 
points  av  av  as,  #4,  of  the  direct  current 
generator  to  four  collector  rings   Dv 
four-phase  converter,  of  the  E.M.F. 


D 


gves  a 


per  phase. 

The  current  per  phase  is  (neglecting  losses  and  phase 
displacement), 


since  the  alternating  power,  2  El  Iv  must  equal  the  direct 
current  power,  E  I. 

Connecting  three  equidistant  points  of  the  commutator 
to  three  collector  rings  as  in  Fig.  109  gives  a  three-phase 
converter. 

In  Fig.  110  the  three  E.M.F.'s  between  the  three  collec- 
tor rings  and  the  neutral  point  of  the  three-phase  system  (or  Y 
voltages)  are  represented  by  the  vectors  OEV  OEV  OEy  thus 
the  E.M.F.  between  the  collector  rings  or  the  delta  voltages 
by  vectors  ~E^Ey  ~EJEV  and  £^EV  The  E.M.F.  ~OE^  is,  how- 
ever, nothing  but  half  the  E.M.F.  El  in  Fig.  107,  of  the 

single-phase    converter,    that    is,    =  -  =..     Hence    the    Y 

2  V2 

voltage  or  voltage  between  collector  ring  and  neutral  point 
of  a  three-phase  converter  is, 


222  ELECTRICAL   ENGINEERING. 

_E 

> 

and  the  delta  voltage  thus, 


2  V2 

Since  the  total  three-phase  power  3  71El  equals  the  total 
continuous  current  power  IE,  it  is, 

IE        2  V2  , 


In  general,    in  an   ;/  phase   converter,  or  converter  in 
which  n  equidistant  points  of  the  commutator  (in  a  bipolar 


Fig.  110. 


machine,  or  n  equidistant  points  per  pair  of  poles,  in  a  multi- 
polar  machine  with  multiple  wound  armature)  are  connected 
to  n  collector  rings,  the  voltage  between  any  collector  ring 
and  the  common  neutral,  or  star  voltage,  is, 


~2V2' 

consequently  the   voltage   between  two   adjacent  collector 
or  ring  voltage,  is, 

E  sin  - 


n          V2 

2?r  . 

since  -  -  is  the  angular  displacement  between  two  adjacent 

collector  rings,  and  herefrom  the  current  per  line  or  star 
current  is  found  as 


SYNCHRONOUS   CONVERTERS.  223 

2V2/ 

7i~  ~ir~' 

and  the  current  flowing  from  line  to  line,  or  from  collector 
ring  to  adjacent  collector  ring,  or  ring  current, 

/'== 


.       7T 

n  X  sin  - 
n 


As  seen  in  the  preceding,  in  the  single-phase  converter 
consisting  of  a  closed-circuit  armature  tapped  at  two  equi- 
distant points  to  the  two  collector  rings,  the  alternating 

voltage  is  — =  times  the  direct  current  voltage,  and  the  alter- 
Y2 

nating  current  V2  times  the  direct  current.  While  such  an 
arrangement  of  single-phase  converter  is  the  simplest,  re- 
quiring only  two  collector  rings,  it  is  undesirable  especially 
for  larger  machines,  on  account  of  the  great  total,  and 
especially  local  PR  heating  in  the  armature  conductors,  as 
will  be  shown  in  the  following,  and  due  to  the  waste  of 
E.M.F.,  since  in  the  circuit  from  collector  ring  to  collector 
ring  the  E.M.F.'s  induced  in  the  coils  next  to  the  leads  are 
wholly  or  almost  wholly  opposite  to  each  other. 

The  arrangement  which  I  have  called  the  "  Two  Circuit 
Single-phase  Converter,"  and  which  is  diagrammatically 


A3' 


Fig.  111. 

shown  in  Fig.  Ill,  is  therefore  preferable.  The  step-down 
transformer  T  contains  two  independent  secondary  coils  A 
and  B,  of  which  the  one  A  feeds  into  the  armature  over 


224  ELECTRICAL   ENGINEERING. 

conductor  rings  Z\  Dz  and  leads  a^  a2,  the  other  B,  over 
collector  rings  Ds  D4  and  leads  az  a4,  so  that  the  two  circuits 
tfjtfg  and  tf8#4are  in  phase  with  each  other,  and  each  spreads 
over  120°  arc  instead  of  180°  arc  as  in  the  single  circuit 
single-phase  converter. 

In  consequence  thereof,  in  the  two  circuit  single-phase 
converter  the  alternating  induced  E.M.F.  bears  to  the  con- 
tinuous current  E.M.F.  the  same  relation  as  in  the  three- 
phase  converter,  that  is, 


2  V2 

and    from   the  equality  of    alternating    and    direct  current 
power, 


it  follows,  that  each  of  the  two  single-phase  supply  currents 
is, 


V3 

As  seen,  in  this  arrangement,  one-third  of  the  armature, 
from  a±  to  as  and  from  a2  to  #4,  carries  the  direct  current 
only,  the  other  two-thirds,  from  a^  to  av  and  from  as  to  #4,  the 
differential  current. 

A  six-phase  converter  is  usually  >  fed  from  a  three-phase 
system  by  three  transformers.  These  transformers  can 
either  have  each  one  secondary  coil  only,  of  twice  the  star 

£ 

or  Y  voltage,  =  —  which  connects  with  its  two  terminals 

V2 

two  collector  rings  leading  to  two  opposite  points  of  the 
armature,  or,  as  usually  preferred,  to  insure  more  uniform 
distribution  of  currents,  each  of  the  step-down  transformers 
contains  two  independent  secondary  coils,  and  each  of  the 
two  sets  of  secondary  coils  is  connected  in  three-phase  delta 
or  Y,  but  the  one  set  of  coils  reversed  with  regard  to  each 
other,  thus  giving  two  three-phase  systems  which  join  to  a 
six-phase  system. 


S  YNCHRONO  US   CONVER  TERS. 


225 


For  further  arrangements  of  six-phase  transformation, 
see  "  Theory  and  Calculation  of  Alternating  Current  Phe- 
nomena," third  edition,  Chapter  XXIX. 

Table  No.  1  gives,  with  the  direct  current  voltage  and 
direct  current  as  unit,  the  alternating  voltages  and  currents 
of  the  different  converters. 


,,; 

M 

H 
Z 

ID 

H  ;/) 

M 

M 

u  2 

X 

u 

X 
cu 

X 
HrD 

«^3 

Js 

E 

li 

S 

X 

H 

sj 

2 

5       C/2 

c  | 

1 

M 

g 

I 

j 

a 

X 

« 

tti 

H 

H 

fa 

U2 

H 

« 

Volts  between  col- 

1 

1 

1 

1 

1 

1 

lector  ring  and 

2  "^2 

2V2 

2  "^2 

2  "^2 

2  "^2 

2  ^2 

neutral  point    . 

=  .354 

=  354 

=  354 

=  354 

=  354 

=  354 

Volts  between  ad- 

1 

_V£ 

^3 

1 

7T 

sin  - 

jacent  collector 

^2 

2  ^2 

2^2 

2^2 

- 

n 

rings   .... 

1.0 

=  .707 

=  612 

=  612 

i^^  .5 

=.354 

.183 

V2 

Amperes  per  line, 

V2 

V2 

2^2 

1 

V2 

~3~ 

2V2 

n 

1.0 

=1.414 

=  817 

=.943 

=  707 

=  472 

.236 

Amperes  between 

V2 

2^2 

V2 

J    « 

adjacent  lines  . 

V2 

V3 

3^3 

3 

=1.414 

=  817 

=  545 

i  =  .5 

=  472 

.455 

l«?i 

These  currents  give  only  the  energy  component  of  alter- 
nating current  corresponding  to  the  direct  current  output. 
Added  thereto  is  the  current  required  to  supply  the  losses 
in  the  machine,  that  is,  to  rotate  it,  and  the  wattless  com- 
ponent if  a  phase  displacement  is  produced  in  the  converter. 


III.  Variation  of  the  Ratio  of  Electro-Motive-Forces. 

The  preceding  ratios  of  E.M.F.'s  apply  strictly  only  to 
the  induced  E.M»F.'s,  and  that  under  the  assumption  of  a 
sine  wave  of  alternating  induced  E.M.F. 

The  latter  is  usually  a  sufficiently  close  approximation, 
since  the  armature  of  the  converter  is  a  multitooth  structure, 
that  is,  contains  a  distributed  winding. 


226  ELECTRICAL    £.i 

The  ratio  between  the  difference  of  potential  at  the  com- 
••>mutator  brushes  and  that  at  the  collector  rings  of  the  con- 
vcr.er  usually  differs  somewhat  from' the  theoretical  ratio, 
due  to  the  E.M.F.  consumed  in  the  converter  armature,  and 
in  machines  converting  from  alternating  to  continuous  cur- 
rent, also  due  to  the  shape  of  impressed  wave. 

When  converting  from  alternating  to  direct  current, 
under  load  the  difference  of  potential  at  the  commutator 
brushes  is  less  than  the  induced  direct  current  E.M.F.,  and 
the  induced  alternating  E.M.F.  less  than  the  impressed,  due 
to  the  voltage  consumed  by  the  armature  resistance. 

If  the  current  in  the  converter  is  in  phase  with  the  im- 
pressed E.M.F.,  armature  self-induction  has  little  effect ; 
but  reduces  the  induced  alternating  E.M.F.  below  the  im- 
pressed with  a  lagging,  and  raises  it  with  a  leading  current, 
in  the  same  way  as  in  a  synchronous  motor. 

Thus  in  general  the  ratio  of  voltages  varies  somewhat 
with  the  load  and  with  the  phase  relation,  and  with  constant 
impressed  alternating  E.M.F.  the  difference  of  potential  at 
the  commutator  brushes  decreases  with  increasing  load, 
decreases  with  decreasing  excitation  (lag)x  and  in  creases  with 
increasing  excitation  (lead). 

When  converting  from  direct  to  alternating  current  the 
reverse  is  the  case. 

The  direct  current  voltage  stands  in  definite  proportion 
only  to  the  maximum  value  of  the  alternating  voltage  (being 
equal  to  twice  the  maximum  star  voltage),  but  to  the  effective 
value  (or  value  read  by  voltmeter)  only  in  so  far  as  the  latter 

depends  upon  the  former,  being  =  — =  maximum  value  with  a 

V2 
sine  wave. 

Thus  with  an  impressed  wave  of  E.M.F.  giving  a  dif- 
ferent ratio  of  maximum  to  effective  value,  the  ratio  between 
direct  current  and  alternating  current  voltage  is  changed  in 
the  same  proportion  as  the  ratio  of  maximum  to  effective. 

Thus,  for  instance,  with  a  flat-topped  wave  of  impressed 


SYNCHRONOUS   CONVERTERS.  227 

E.M.F.,  the  maximum  value  of  alternating  impressed  E.M.F. 
and  thus  the  direct  current  voltage  depending  thereupon, 
are  lower  than  with  a  sine  wave  of  the  same  effective  value,  « 
while  with  a  peaked  wave   of   impressed   E.M.F.  they  are 
higher,  by  as  much  as  10%  in  extreme  cases. 

In  determining  the  wave  shape  of  impressed  E.M.F. 
at  the  converter  terminals,  not  only  the  wave  of  induced 
generator  E.M.F.,  but  also  that  of  the  converter  induced 
E.M.F.,  or  counter  E.M.F.,  may  be  instrumental.  Thus, 
with  a  converter  connected  directly  to  a  generating  system 
of  very  large  capacity,  the  impressed  E.M.F.  wave  will  be 
practically  identical  with  the  generator  wave,  while  at  the 
terminals  of  a  converter  connected  to  the  generator  over 
long  lines  with  reactive  coils  and  inductive  regulators  inter- 
posed, the  wave  of  impressed  E.M.F.  may  be  so  far  modi- 
fied by  that  of  the  counter  E.M.F.  of  the  converter,  as 
to  resemble  the  latter  much  more  than  the  generator  wave, 
and  thereby  the  ratio  of  conversion  may  be  quite  different 
from  that  corresponding  to  the  generator  wave. 

Furthermore,  for  instance,  in  three-phase  converters  fed 
by  ring  or  delta  connected  transformers,  the  Y  or  star 
E.M.F.  at  the  converter  terminals,  which  determines  the 
direct  current  voltage,  may  differ  from  the  Y  or  star 
E.M.F.  impressed  by  the  generator,  by  containing  different 
third  and  ninth  harmonics,  which  cancel  when  compound- 
ing the  Y  voltages  to  the  delta  voltage,  and  give  identical 
delta  voltages,  as  required. 

Hence,  the  ratios  of  E.M.F.'s  given  in  paragraph  2  have 
to  be  corrected  by  the  drop  of  voltage  in  the  armature,  and 
have  to  be  multiplied  by  a  factor  which  is  V2  times  the  ratio 
of  effective  to  maximum  value  of  impressed  wave  of  star 
E.M.F.  (V2  being  the  ratio  of  maximum  to  effective  of  the 
sine  wave  on  which  the  ratios  in  paragraph  2  were  based). 

With  an  impressed  wave  differing  from  sine  shape,  a 
current  of  higher  frequency,  but  generally  of  negligible 
magnitude,  flows  through  the  converter  armature,  due  to 


228 


ELECTRICAL   ENGINEERING. 


the    difference    between    impressed    and    counter    E.M.  F. 
wave. 

IV.   Armature  Current  and  Heating. 

The  current  flowing  in  the  armature  conductors  of  a 
converter  is  the  difference  between  the  alternating  current 
input  and  the  direct  current  output. 

In  Fig.  112,  av  av  are  two  adjacent  leads  connected  with 
the  collector  rings  Dv  D2  in  an  ;/-phase  converter.  The 

alternating  E.M.F.  between  a^ 
and  av  and  thus  the  alter- 
nating energy  current  flowing 
in  the  armature  section  be- 
tween a1  and  av  will  reach  a 
maximum  when  this  section  is 
midway  between  the  brushes 
BI  and  Bv  as1  shown  in  Fig. 
112. 

The  direct  current  in  every  armature  coil  reverses  at 
the  moment  when  the  the  coil  passes  under  brush  Bl  or 
By  thus  is  a  rectangular  alternating  current  as  shown  in 
Fig.  113  as  /.  At  the  moment  when  the  alternating  energy 


Fig.  113. 


current  is  a  maximum,  an  armature  coil  d  midway  between 
two  adjacent  alternating  leads  a^  and  az  is  midway  between 
the  brushes  Bt  and  By  as  in  Fig.  112,  and  is  thus  in  the 
middle  of  its  rectangular  continuous  current  wave,  and  con- 
sequently in  this  coil  the  alternating  energy  and  the  rec- 
tangular direct  current  are  in  phase  with  each  other,  but 


.5" YNCHRONO  US   CONVER TERS. 


229 


opposite,  as  shown  in  Fig.  113  as  /x  and  /,  and  the  actual 
current  is  their  difference,  as  shown  in  Fig.  114. 

In  successive  armature  coils,  the  direct  current  reverses 
successively.  That  is,  the  rectangular  currents  flowing  in 
successive  armature  coils  are  successively  displaced  in  phase 


Fig.  114. 

from  each  other;  and  since  the  alternating  current  is  the 
same  in  the  whole  section  a^  av  and  in  phase  with  the  rec- 
tangular current  in  the  coil  d,  it  becomes  more  and  more 
out  of  phase  therewith  when  passing  from  coil  d  towards  a^ 
or  av  as  shown  in  Figs.  115,  116,  117,  and  118,  until  the 
maximum  phase  displacement  between  alternating  and  rec- 
tangular current  is  reached  at  the  alternating  leads  a^  and 

av  and  is  equal  to  -• 
n 

Thus,  if  E  —  direct  voltage,  and  /  =  direct  current,  in 
an  armature  coil  displaced  by  angle  <o  from  the  position  d, 
midway  between  two  adjacent  leads  of  the  ^-phase  converter, 

the  direct  current  is  -  fcr  the  half  period  from  0  to  IT,  and 
the  alternating  current  is, 


where, 


V2/'  sin  (<£  -  o>), 
/V2 


7T 

#  sin - 
n 


is  the  effective  value  of  the  alternating  current.     Thus,  the 
actual  current  in  this  armature  coil  is, 

jQ  =  ^/2  I'  sin  (<£  —  o>)  —  ^ 
=  /r4sinQ-<o)      ^\ 


.       7T 

wsm  - 
n 


230  ELECTRICAL   ENGINEERING. 

and  its  effective  value, 


;/  sin  - 
n 


.          8  16coso> 

=  ~9  V  ~  

^     »  .        7T  7T 


778. 


SYNCHRONOUS   CONVERTERS.  231 


Since  ^  is  the  current  in  the  armature  coil  of  a  direct  cur- 
rent generator  of  the  same  output,  we  have, 


the  ratio  of  the  energy  loss  in  the  armature  coil  resistance  of 
the  converter,  to  that  of  the  direct  current  generator  of  the 
same  output,  and  thus  the  ratio  of  coil  heating. 

This  ratio  is  a  maximum  at  the  position  of  the  alternat- 
ing leads,  w  =  - ,  and  is, 

16  cos- 


.     07T  .       7T 

z  sm2  -  HIT  sm  - 

72  // 


It  is   a  minimum  for  a  coil  midway   between   adjacent 
alternating  leads,  cu  =  0,  and  is, 

*-- ^+i-  16 


7T  .       7T 

/2Z  sin2  —  mr  sm  — 

;/  /^ 

Integrating  over  <o  from  0  (coil  d]  to  -,  that  is,  over  the 
whole  phase  or  section  al  ay  we  have, 

8  16 


TTjQ  „      .     2  7T 

?r  sin2  - 


the  ratio  of  the  total  energy  loss  in  the  armature  resistance 
of  an  «-phase  converter  to  that  of  the  same  machine  as 
direct  current  generator  at  the  same  output,  or  the  relative 
armature  heating. 

Thus,  to  get  the  same  energy  loss  in  the  armature  con- 
ductors, and  consequently  the  same  heating  of  the  armature, 
the  current  in  the  converter,  and  thus  its  output,  can  be 


232  ELECTRICAL   ENGINEERING. 

increased  in  the  proportion  —  over  that  of   the  direct    cur- 

rent generator. 

The  calculation  for  the  two  circuit  single-phase  converter 
is  somewhat  different  ;  since  in  this  in  one-third  of  the  arma- 
ture the  PR  loss  is  that  of  the  direct  current  output,  and 

27T 

only  in  the  other  two-thirds,  —  or  as  arc  —  ,  —  the  differen- 

o    • 

tial  current  flows.  Thus  in  an  armature  coil  displaced  by 
angle  <o  from  the  center  of  this  latter  section,  the  resultant 
current  is, 


giving  the  effective  value, 


thus  the  relative  heating, 

Vu>    = 


with  the  minimum  value,  w  =  0 
11        16 


and  the  maximum  value,  w  =  -, 

o 

11         8 

*»=y-.v§=2-18' 

and   the    average    current    heating   in    two-thirds    of    the 
armature, 

3   F5              H        16     •    »• 
Th  =  -    I     yw  d<*  — —  sin  - 

/        '  Q  ^/Q  ^ 

7T  '^O  O  7T    VO  O 


S  YNCHR  ONO  US    CONVER  TERS. 


233 


in  the  remaining  third  of   the   armature,  I\  =  1,  thus  the 
average, 


and  thus  the  rating, 


3 

=  1.072, 


By  substituting  for  ;/  in  the  general  equations  of  cur- 
rent heating  and  rating  based  thereon,  numerical  values,  we 
get  thus  the  table, 


o 

"  9 

M 

.     -f. 

M 

H   ^ 

u  < 

s  < 

c/; 

<> 

Si    I 

S  X 

TYPE. 

ill 

GLE  Cl 
NGLE-P 

U  a. 

2  w 
ij 

f 

H 

x 
t> 

M 

X 
E 

a, 
A 

j 

M 

M 

'f. 

x 

0. 

Q 

" 

Sw 

B 

k, 

w 

H 

8 

n       

2 

2 

3 

4 

6 

12 

7o      LOO 

.45 

.70 

.225 

.20 

.19 

.187 

\ 

7,/t     

1.00 

3.00 

2.18 

1.20 

.73 

.42 

.24 

£.187 

r    

1.00 

1.37 

1.072 

.555 

.37 

.26 

.20 

Rating  (by  mean 
arm.  heating), 

1.00 

.85 

.97 

1.34 

1.64 

1.96 

2.24 

2.31 

As  seen,  in  the  two  circuit  three-phase  converter  the 
armature  heating  is  less,  and  more  uniformly  distributed, 
than  in  the  single  circuit  single-phase  converter. 

In  these  values,  the  small  energy  current  supplying  the 
losses  in  the  converter  is  neglected. 

Assuming  this  current  =  ql',  where  /'  =  energy  current 
corresponding  to  direct  current  (where  q=  .04  when  assum- 
ing 4%  loss  in  mechanical  and  molecular  magnetic  friction), 
the  total  alternating  energy  current  is, 

//  =  /'(!+?), 
and  the  resultant  current, 

*/  =  /'  sin  (A  -  ^  -I-  =  I-(  4a  +  y)sm(0  -_o>)  _  ^  J 

' 


7T 

n  sin  - 
n 


234  ELECTRICAL   ENGINEERING. 

thus  its  effective  value, 


74  /8(1+?)2  ~          16(1 

oV-          — ' --T 


«2  sin2  -  tiTT  sin  - 

n  » 

hence, 

/o'      8  (1  +  qf1  16  (1  +  g)  cos  co 

-          /z2  sin2  -  KTT  sin  - 

2  »  H 

(        8  8  cos  co 

=  >  +  2?\- 


Integrated  over  w  from  0  to  - 


In  the  two  circuit  single-phase  converter,  we  have, 
/'  =  /'(1  +  q)  sin  (</>  -co)- 


+  ^)  cos  co 


r_  8  (1  +  ^)2  +  3  _  16(1  +  ?)  cos <o 
=  ~          3  w  V3 


3 

2JV 
3 


SYNCHRONOUS    CONVERTERS. 


235 


Herefrom  we .  get  the  values  for  the  relative  current 
heating  r",  and  the  corresponding  rating  — — ,  by  assuming 
q  =  .04. 


z 

|a 

H 

H 

g 

*  £ 

u  *• 

• 

H 

X 

TYPE. 

3 

O  M 

«  « 

E 

X 

1 

u 

HU 

K   J 

J^    J 

• 

E 

jj 

5 

1 

|l 

6  z 

•  g 

M 

D 

>< 

M 

x 

K 

q 

w 

H^ 

H 

fc 

0. 

H 

8 

r  -  

1.00 

1.37 

1.072 

.555 

.37 

.20 

.20 

.187 

r  =.    .    . 

1.00 

1.475 

1.149 

.585 

.385 

.267 

.201 

.187 

Rating      .     .     . 

1.00 

.825 

.933 

1.31 

1.61 

1.94 

2.24 

2.31 

If  a  phase  displacement  exists  between  current  and 
E.M.F.  in  the  converter,  the  current  can  be  resolved  into 
an  energy  component  in  phase  with  the  E.M.F.  and  a  watt- 
less component  in  quadrature  therewith. 

The  loss  of  power  in  a  resistance  due  to  an  alternating 
current  is  equal  to  the  sum  of  the  losses  due  to  the  two 
components,  the  energy  current  and  the  wattless  current ; 
and  thus  if  a  phase  displacement  is  produced  in  a  converter, 
the  loss  of  power  due  to  the  wattless  current  has  to  be 
added  to  the  loss  of  power  due  to  the  energy  current,  as 
discussed  and  recorded  in  above-given  tables. 

Assuming,  for  instance,  a  wattless  current  at  full  load 
(leading 'or  lagging)  of  30%  of  full-load  current,  the  addi- 
tional energy  loss  in  the  armature  corresponding  thereto 
amounts  to  .32  =  .09  of  that  of  the  same  machine  as  alter- 
nating current  generator  or  synchronous  machine. 

The  relative  IZR  losses  in  the  armature  of  a  direct 
current  machine  and  the  different  types  of  synchronous 
machines,  at  the  same  power  generated,  viz.,  consumed  in 
the  armature,  are  calculated  in  the  following  table. 

Armature  T"R  of  Commutating  and  of  Synchronous 
Machines,  at  equal  electric  power. 

Resistance  from  brush  to  brush  =  1. 

Total  current  in  direct  current  machine  =  1. 


236 


ELECTRICAL   ENGINEERING. 


h 
Z; 

M 

« 

M 

•y. 

8 

K 

u  < 

X 

*   X 

~   X 

•f. 

X 

„ 

D 

U  i 

u  "f 
x  K 

X 

uT 

u^ 

w  >J 

&  u 

X 

> 

V: 

M 

as 

•j  z 

Iw 

o 

X 

M 
p 

h 

Q 

C/3 

H 

H 

CO 

H 

S 

Number    of 

circuits     . 

2 

2 

2 

3 

4 

6 

12 

w 

Resistances 

4 

4 

2 

1 

4 

per  circuit 

o 

2 

3 

3 

1 

3 

3 

Current  per 
circuit  .     . 

.5 

V2 

V2 

2V2 

5 

V2 

455 

V2        V^ 

2 

V3 

3^3 

3 

«  sin  - 

n 

Total  I^R 

9 

16 

32 

8 

8        8 

9 

—  _  —  1.185 

2t 

--.889 

;/'2sin-~z    «* 

.09/27?   .     . 

.09 

.18 

.16 

.106 

.09 

.08 

.074 

.073 

Adding  these  values  to  the  values  given  in  above  tables, 
gives  a  relative  heating  of  the  different  types  of  machines, 
and  permissible  rating  (as  based  on  armature  heating)  as 
follows : 

30%   WATTLESS    CURRENT. 


H 

Sti 

H 

H 

Z 

8  < 

M 

tf. 

OS 

5g 

S  K 

™    v 

5 

TYPE. 

sa 

^3 

J  O 

II 

1 

H 
OS 

X 

E 

M 
§ 

Efl 

i 

X 

t; 
•f. 

gj 

X 

PM 

Q 

« 

HW 

h 

a; 

8 

r'=  

1.00 

1.475 

1.149 

.585 

.385 

.^67 

.201 

.187 

r  "^~ 

1  00 

1655 

1255 

691 

475 

347 

275 

260 

Ratir.g      .     .     . 

1.00 

.78 

.89 

1.20 

1.45 

1.70 

1.91 

1.90 

V.   Armature   Reaction. 

The  armature  reaction  of  the  polyphase  converter  is 
the  resultant  of  the  armature  reactions  of  the  machine  as 
direct  current  generator  and  as  synchronous  motor.  If  the 
commutator  brushes  are  set  at  right  angles  to  the  field  poles 


SYNCHRONOUS   CONVERTERS.  237 

or  without  lead  or  lag,  as  is  usually  done  in  converters,  the 
direct  current  armature  reaction  consists  in  a  polarization 
in  quadrature  behind  the  field  magnetism.  The  armature 
reaction  due  to  the  energy  component  of  the  alternating 
current  in  a  synchronous  motor  consists  of  a  polarization  in 
quadrature  ahead  of  the  field  magnetism,  which  is  opposite 
to  the  armature  reaction  as  direct  current  generator. 

Let  m  =  total  number  of  turns  on  the  bipolar  armature 
of  an  n  =  phase  converter,  /  =  direct  current,  then  then  um- 
ber of  turns  in  series  between  the  brushes  =  — ,  hence  the 

total  armature  ampere  turns,  or  polarization  =  ^- .     Since, 

however,  these  ampere  turns  are  not  unidirectional,  but 
distributed  over  the  whole  surface  of  the  armature,  their 
resultant  is, 

ml  (+T 

9  =  -~-  avg.  cos  < 

£i  '  ^_ 

2 

And  since 

( "*""*      2 
avg.  cos.  )       =  — . 

we  have,  SF  =  -  -  =  direct  current  polarization  of  the  con- 
verter (or  direct  current  generator)  armature. 

In   an    n  =  phase  converter  the  number  of   turns  per 

phase  =  — -  The  current  per  phase  or  current  between 
two  adjacent  leads  (ring  current)  is, 

/_.    V27 


/' = 


.       7T 

n  sin  — 
n 


hence  ampere  turns  per  phase, 


238  ELECTRICAL   ENGINEERING. 

: 

These  ampere  turns  are  distributed  over  -  of  the  circum 
ference  of  the  armature,  and  their  resultant  is  thus, 


and  since, 


ml' 
=  —  avg.  cos 


(     «       n    .    TT 
av£.  cos.  J        =  -  sm  - 

n  "  H 


We  have, 

~  _  V2  ml 

«M  —  —  =  resultant  polarization, 

in  effective  ampere  turns,  of  one  phase  of  the  converter. 

The  resultant  M.M.F.  of  n  equal  M.M.F.'s  of  effective 
value   of  9^,  thus   maximum  value  of  9^  V2,  acting  under 

equal  angles  — >  and  displaced  in  phase  from  each  other  by  - 


*  z, 


of  a  period,  or  phase  angle  —  ,  if  found  thus  : 


Let  (F,-  = 


r-        /          2/ir\ 
.  —  [Fv 2  sin  I  <f>  —  =  one  of  the  M.M.F.s  of  phase 

2/tr  */  2/V 

angle  A= ,  acting  in  the  direction  w  =  —  —  • 

n  n 

That  is,  the  zero  point  of  one  of  the  M.M.F.'s  Syis  taken  as 
zero  point  of  time  <j>,  and  the  direction  of  this  M.M.F.  as 
zero  point  of  direction  o>. 

The  resultant  M.M.F.  in  any  direction  w  is  thus, 

XI        2  /V\ 
r'SFicbsfa  H J 


.  /  4/ 

^  sm  ( <^>  +  w 1  +  »  sin  (<£  —  w) 


f 

SYNCHRONOUS   CONVERTERS.  239 

and  since, 

n 

^i  sin  f  (f)  -h  to  --  -j  =  0, 

we  have, 

•SF:^—^  sin  (*-<»); 

that  is, 

The  resultant  M.M.F.  in  any  direction  to  has  the  phase, 

<f>  =  to, 
and  the  intensity, 


thus  revolves  in  space  with  uniform  velocity  and  con- 
stant intensity,  in  synchronism  with  the  frequency  of  the 
alternating  current. 

Since  in  the  converter, 


we  have 


the  resultant  M.M.F.  of  the  alternating  energy  current  in 
the  n  —  phase  converter. 

This  M.M.F.  revolves  synchronously  in  the  armature  of 
the  converter;  and  since  the  armature  rotates  at  synchronism, 
the  resultant  M.M.F.  stands  still  in.  space,  or  with  regard 
to  the  field  poles,  in  opposition  to  the  direct  current  polariza- 
tion. Since  it  is  equal  thereto,  it  follows,  that  the  resultant 
armature  reactions  of  the  direct  current  and  of  the  cor- 
responding alternating  energy  current  in  the  synchronous 
converter  are  equal  and  opposite,  thus  neutralize  each  other, 
and  the  resultant  armature  polarization  equals  zero.  The 
same  is  obviously  the  case  in  an  "  Inverted  Converter  "  ; 
that  is,  a  machine  changing  from  direct  to  alternating 
current. 

The  conditions-  in  a  single-phase  converter  are  different, 


240  ELECTRICAL   ENGINEERING. 

however.  At  the  moment  when  the  alternating  current  =  0, 
the  full  direct  current  reaction  exists.  At  the  moment 
when  the  alternating  current  is  a  maximum,  the  reaction  is 
the  difference  between  that  of  the  alternating  and  of  the 
direct  current  ;  and  since  the  maximum  alternating  current 
in  the  single-phase  converter  equals  twice  the  direct  current, 
at  this  moment  the  resultant  armature  reaction  is  equal  but 
opposite  to  the  direct  current  reaction. 

Hence,  the  armature  reaction  oscillates  with  twice  the 
frequency  of  the 'alternating  current,  and  with  full  intensity, 
and  since  it  is  in  quadrature  with  the  field  excitation,  tends 
to  shift  the  magnetic  flux  rapidly  across  the  field  poles,  and 
thereby  tends  to  cause  sparking  and  energy  losses.  This 
oscillating  reaction  is,  however,  essentially  reduced  by  the 
damping  effect  of  the  magnetic  field  structure. 

Since  in  consequence  hereof  the  commutation  of  the 
single-phase  converter  is  not  as  good  as  of  the  polyphase 
converter,  in  the  former  usually  voltage  commutation  has  to 
be  resorted  to ;  that  is,  the  brushes  shifted  from  the  position 
midway  between  the  field  poles  ;  and  in  consequence  thereof, 
the  continuous  current  ampere  turns  inclosed  by  twice  the 
angle  of  lead  of  the  brushes  act  as  a  demagnetizing  armature 
reaction,  and  require  a  corresponding  increase  of  the  field 
excitation  under  load. 

Since  the  resultant  main  armature  reactions  neutralize 
each  other  in  the  polyphase  converter,  there  remain  only  — 

1st.  The  armature  reaction  due  to  the  small  energy 
current  required  to  rotate  the  machine ;  that  is,  to  cover  the 
internal  losses  of  power,  which  is  in  quadrature  with  the 
field  excitation  or  distorting,  but  of  negligible  magnitude. 

2d.  The  armature  reaction  due  to  the  wattless  com- 
ponent of  alternating  current  where  such  exists,  and 

3d.  An  effect  of  oscillating  nature,  which  may  be  called 
a  "  Higher  Harmonic  of  Armature  Reaction." 

The  direct  current,  as  rectangular  alternating  current  in 
the  armature,  changes  in  phase  from  coil  to  coil,  while  the 


SY'NCHONROUS   CONVERTERS.  241 

alternating  current  is  the  same  in  a  whole  section  of  the 
armature  between  adjacent  leads. 

Thus  while  the  resultant  reactions  neutralize,  a  local 
effect  remains  which  in  its  relation  to  the  magnetic  field 
oscillates  with  a  period  equal  to  the  time  of  motion  of  the 
armature  through  the  angle  between  adjacent  alternating 
leads ;  that  is,  double  frequency  in  a  single-phase  converter 
(in  which  it  is  equal  in  magnitude  to  the  direct  current 
reaction,  and  is  the  oscillating  armature  reaction  discussed 
above),  triple  frequency  in  a  three-phase  converter,  and 
quadruple  frequency  in  a  four-phase  converter. 

The  amplitude  of  this  oscillation  in  a  polyphase  con- 
verter is  small,  and  its  influence  upon  the  magnetic  field  is 
usually  negligible,  due  to  the  damping  effect  of  the  field 
spools,  which  act  like  a  short  circuited  winding  for  an  oscil- 
lation of  magnetism. 

A  polyphase  converter  on  unbalanced  circuit  can  be  con- 
sidered as  a  combination  of  a  balanced  polyphase  and  a 
single-phase  converter ;  and  since  even  single-phase  con- 
verters operate  quite  satisfactorily,  the  effect  of  unbalanced 
circuits  on  the  polyphase  converter  is  comparatively  small, 
within  reasonable  limits. 

Since  the  armature  reaction  of  the  direct  current  and 
of  the  alternating  current  in  the  converter  neutralize  each 
other,  no  change  of  field  excitation  is  required  in  the  con- 
verter with  changes  of  load. 

Furthermore,  while  in  a  direct  current  generator  the 
armature  reaction  at  given  field  strength  is  limited  by  the 
distortion  of  the  field  caused  thereby,  this  limitation  does 
not  exist  in  a  converter;  and  a  much  greater  armature 
reaction  can  be  safely  used  in  converters  than  in  direct 
current  generators,  the  distortion  being  absent  in  the  former. 

Since  the  armature  heating  is  relatively  small,  the  prac- 
tical limit  of  overload  capacity  of  a  converter  is  that  due  to 
the  commutator,  and  is  usually  far  higher  than  in  a  direct 
current  generator,  since  the  distortion  of  field,  which  causes 


242  ELECTRICAL   ENGINEERING. 

sparking  on   the  commutator  under  overloads    in  a  direct 
current  generator,  is  absent  in  a  converter. 

The  theoretical  limit  of  overload  —  that  is,  the  overload 
at  which  the  converter  as  synchronous  motor  drops  out  of 
step  and  comes  to  a  standstill  —  is  far  beyond  reach  at  steady 
frequency  and  constant  impressed  alternating  voltage,  while 
on  an  alternating  circuit  of  pulsating  frequency  or  drooping 
voltage  it  obviously  depends  upon  the  amplitude  and  period 
of  the  pulsation  of  frequency  and  on  the  drop  of  voltage. 

VI.  Wattless  Currents  and  Compounding. 

Since  the  polarization  due  to  the  alternating  energy  cur- 
rent as  synchronous  motor  is  in  quadrature  ahead  of  the  field 
magnetization,  the  polarization  or  magnetizing  effect  of  the 
lagging  component  of  alternating  current  is  in  phase,  that  of 
the  leading  component  of  alternating  current  in  opposition  to 
the  field  magnetization.  That  is,  in  the  converter  no  mag- 
netic distortion  exists,  and  no  armature  reaction  at  all  if  the 
current  is  in  phase  with  the  impressed  E.M.F.,  while  the  arma- 
ture reaction  is  demagnetizing  with  a  leading,  magnetizing 
with  a  lagging  current. 

Thus  if  the  alternating  current  is  lagging,  the  field  ex- 
citation at  the  same  impressed  E.M.F.  has  to  be  lower,  and 
if  the  alternating  current -is  leading,  the  field  excitation  has 
to  be  higher,  than  required  with  the  alternating  current  in 
phase  with  the  E.M.F.  Inversely,  by  raising  the  field  ex- 
citation a  leading  current,  or  by  lowering  it  a  lagging  current, 
can  be  produced  in  a  converter  (and  in  a  synchronous  motor). 

Since  the  alternating  current  can  be  made  magnetizing 
or  demagnetizing  according  to  the  field  excitation,  at  con- 
stant impressed  alternating  voltage,  the  field  excitation  of 
the  converter  can  be  varied  through  a  wide  range  without 
noticeably  affecting  the  voltage  at  the  commutator  brushes  ; 
and  in  converters  of  high  armature  reaction  and  relatively 
weak  field  full-load  and  over-load  can  be  carried  by  the 


SYNCHRONOUS   CONVERTERS.  243 

machine  without  any  field  excitation  whatever,  that  is,  by 
exciting  the  field  by  armature  reaction  by  the  lagging 
alternating  current.  Such  converters  without  field  excita- 
tion, or  "  Reaction  Converters,"  must  always  run  with  more 
or  less  lagging  current,  that  is,  give  the  same  reaction  en 
the  line  as  induction  motors,  which,  as  known,  are  far  more 
objectionable  than  synchronous  motors  in  their  reaction  on 
the  alternating  system. 

Conversely,  however,  at  constant  impressed  alternating 
voltage  the  direct  current  voltage  of  a  converter  cannot  be 
varied  by  varying  the  field  excitation  (except  by  the  very 
small  amount  due  to  the  change  of  the  ratio  of  conversion), 
but  a  change  of  field  excitation  merely  produces  wattless 
currents,  lagging  or  magnetizing  with  a  decrease,  leading  or 
demagnetizing  with  an  increase  of  field  excitation.  Thus  te 
vary  the  continuous  current  voltage  of  a  converter,  the  im- 
pressed alternating  voltage  has  to  be  varied.  This  can  be 
done  either  by  potential  regulator  or  compensator,  that  is, 
transformers  of  variable  ratio  of  transformation,  or  by  the 
effect  of  wattless  currents  on  self-induction.  The  latter 
method  is  especially  suited  for  converters,  due  to  their  ability 
of  producing  wattless  currents  by  change  of  field  excitation. 

The  E.M.F.  of  self-induction  lags  90°  behind  the  current. 
Thus  if  the  current  is  Tagging  90°  behind  the  impressed 
E.M.F.,  the  E.M.F.  of  self-induction  is  180°  behind,  or  in 
opposition  to  the  impressed  E.M.F.,  and  thus  reduces  it.  If 
the  current  is  90°  ahead  of  the  E.M.F.,  the  E.M.F.  of  self- 
induction  is  in  phase  with  the  impressed  E.M.F.,  thus  adds 
itself  thereto  and  raises  it.  Thus  if  self-induction  is  inserted 
into  the  lines  between  converter  and  constant  potential  gen- 
erator, and  a  wattless  lagging  current  is  produced  by  the 
converter  by  a  decrease  of  its  field  excitation,  the  E.M.F.  of 
self-induction  of  this  lagging  current  in  the  line  lowers  the 
alternating  impressed  voltage  at  the  converter  and  thus  its 
direct  current  voltage ;  and  if  a  -wattless  leading  current  is 
produced  by  the  converter  by  an  increase  of  its  field  excita- 


244  ELECTRICAL    ENGINEERING. 

tion,  the  E.M.F,  of  self-induction  of  this  leading  current 
raises  the  impressed  alternating  voltage  at  the  converter  and 
thus  its  direct  current  voltage. 

In  this  manner,  by  self-induction  in  the  lines  leading  to 
the  converter,  its  voltage  can  be  varied  by  a  change  of  field 
excitation,  or  conversely  its  voltage  maintained  constant  at 
constant  generator  voltage  or  even  constant  generater  excita- 
tion, with  increasing  load  and  thus  increasing  resistance  drop 
in  the  line  ;  or  the  voltage  can  even  be  increased  with  in- 
creasing load,  that  is,  the  system  over-compounded. 

The  change  of  field  excitation  of  the  converter  with 
changes  of  load  can  be  made  automatic  by  tho,  combination 
of  shunt  and  series  field,  and  in  this  manner  a  converter  can 
be  compounded  or  even  over-compounded  similarly  to  a 
direct  current  generator.  While  the  effect  is  the  same,  the 
action,  however,  is  different ;  and  the  compounding  takes 
place  not  in  the  machine  as  with  a  direct  current  generator, 
but  in  the  alternating  lines  leading  to  the  machine,  in  which 
self-induction  becomes  essential. 

VII.    Starting. 

The  polyphase  converter  is  self-starting  from  rest ;  that 
is,  when  connected  across  the  polyphase  circuit  it  starts,  ac- 
celerates, and  runs  up  to  complete  synchronism.  The 
E.M.F.  between  the  commutator  brushes  is  alternating  in 
starting,  with  the  frequency  of  slip  below  synchronism. 
Thus  a  direct  current  voltmeter  or  incandescent  lamps  con- 
nected across  the  commutator  brushes  indicate  by  their 
beats  the  approach  of  the  converter  to  synchronism.  When 
starting,  the  field  circuit  of  the  converter  has  to  be  opened 
or  at  least  greatly  weakened.  The  starting  of  the  polyphase 
converter  is  essentially  a  hysteresis  effect  and  entirely  so  in 
machines  with  laminated  field  poles,  while  in  machines  with 
solid  magnet  poles,  induced  currents  in  the  latter  contribute 
to  the  starting  torque,  but  at  the  same  time  reduce  the  mag- 


SYNCHRONOUS   CONVERTERS,  245 

netic  starting  flux  by  their  demagnetizing  effect.  The  torque 
is  produced  by  the  attraction  between  the  alternating  cur- 
rents of  the  successive  phases  upon  the  remanent  magnetism 
or  induced  currents  produced  by  the  preceding  phase.  It 
necessarily  is  comparatively  weak,  and  from  full-load  to  twice 
full-load  current  is  required  to  start  from  rest  without  load. 

Obviously,  the  single-phase  converter  is  not  self-starting. 
At  the  moment  of  starting,  the  field  circuit  of  the  converter 
is  in  the  position  of  a  secondary  to  the  armature  circuit  as 
primary;  and  since  in  general  the  number  of  field  turns  is 
very  much  larger  than  the  number  of  armature  turns,  ex- 
cessive E.JVLF.'s  may  be  induced  in  the  field,  reaching 
frequently  4000  to  6000  volts,  which  have  to  be  taken  care 
of  by  some  means,  as  by  breaking  the  field  circuit  into 
sections.  As  soon  as  synchronism  is  reached,  which  usually 
takes  from  a  few  seconds  to  a  minute  or  more,  the  field 
circuit  is  closed,  and  the  load  put  on  the  converter.  Obvi-* 
ously,  while  starting,  the  direct  current  side  of  the  converter1 
must  be  open  circuited,  since  the  E.M.F.  between  commu- 
tator brushes  is  alternating  until  synchronism  is  reached. 

When  starting  from  the  alternating  side,  the  converter 
can  drop  into  synchronism  at  either  polarity;  but  its  polarity 
can  be  reversed  by  strongly  exciting  the  field  in  the  right 
direction  by  some  outside  source,  as  another  converter,  etc. 

Since  when  starting  from  the  alternating  side  the  con- 
verter requires  a  very  large  and,  at  the  same  time,  lagging 
current,  it  is  preferable  wherever  feasible  to  start  it  from 
the  direct  current  side  as  direct  current  motor.  This  can 
be  done  when  connected  to  storage  battery  or  direct  current 
generator.  When  feeding  into  a  direct  current  system 
together  with  other  converters  or  converter  stations,  all  but 
the  first  converter  can  be  started  from  the  continuous  cur- 
rent side  by  means  of  rheostats  inserted  into  the  armature 
circuit. 

To  avoid  the  necessity  of  synchronizing  the  converter, 
by  phase  lamps,  with  the  alternating  system  -  in  case  of 


246  ELECTRICAL   ENGINEERING. 

starting  by  direct  current  (which  operation  may  be  difficult 
where  the  direct  current  voltage  fluctuates,  owing  to  heavy 
fluctuations  of  load,  as  railway  systems),  it  is  frequently 
preferable  to  run  the  converter  up  to  or  beyond  synchronism 
by  direct  current ;  then  cut  off  from  the  direct  current,, 
open  the  field,  and  connect  it  to  the  alternating  system,, 
thus  bringing  it  into  step  by  alternating  current. 

VIII.   Inverted  Converters. 

Converters  may  be  used  to  change  either  from  alter- 
nating to  direct  current  or  as  "Inverted  Converters"  from 
direct  to  alternating  current.  While  the  former  use  is  by 
far  more  frequent,  sometimes  inverted  converters  are  de- 
sirable. Thus  in  low-tension  direct  current  systems,  an 
outlying  district  may  be  supplied  by  converting  from  direct 
to  alternating,  transmitting  as  alternating,  and  then  recon- 
verting to  direct  current.  Or  in  a  station  containing  direct 
current  generators  for  short  distance  supply  and  alternatcrs- 
for  long  distance  supply,  the  converter  may  be  used  as  the 
connecting  link  to  shift  the  load  from  the  direct  to  the 
alternating  generators,  or  inversely,  and  thus  be  operated 
either  way  according  to  the  distribution  of  load  on  the 
system. 

When  converting  from  alternating  to  direct  current,  the 
speed  of  the  converter  is  rigidly  fixed  by  the  frequency, 
and  cannot  be  varied  by  its  field  excitation,  the  variation  of 
the  latter  merely  changing  the  phase  relation  of  the  alter- 
nating current.  When  converting,  however,  from  direct  to 
alternating  current  as  only  source  of  alternating  current, 
that  is,  not  running  in  multiple  with  engine  or  turbine 
driven  alternating  current  generators,  the  speed  of  the  con- 
verter as  direct  current  motor  depends  upon  the  field 
strength,  thus  it  increases  with  decreasing  and  decreases 
with  increasing  field  strength.  As  alternating  current  gen- 
erator, however,  the  field  strength  depends  upon  the  intensity 


SYNCHRONOUS   CONVERTERS.  247 

and  phase  relation  of  the  alternating  current,  lagging  current 
reducing  the  field  strength,  and  thus  increasing  speed  and      ^^ 
frequency,  and  leading  current  increasing  the  field  strength,      * 
and  thus  decreasing  speed  and  frequency. 

Thus,  if  a  load  of  lagging  current  is  put  on  an  inverted 
converter,  as,  for  instance,  by  starting  a  converter  thereby 
from  the  alternating  side,  the  demagnetizing  effect  of  the 
alternating  current  reduces  the  field  strength,  and  causes 
the  converter  to  increase  in  speed  and  frequency.  An  in- 
crease of  frequency,  however,  may  increase  the  lag  of  the  . 
current,  and  thus  its  demagnetizing  effect,  and  thereby  still  ^r 
further  increase  the  speed,  so  that  the  acceleration  may 
become  so  rapid  as  to  be  beyond  control  by  the  field  rheostat, 
and  endanger  the  machine.  Hence,  inverted  converters 
have  to  be  carefully  watched,  especially  when  starting  other 
converters  from  them  ;  and  some  absolutely  positive  device 
is  necessary  to  cut  the  inverted  converter  off  the  circuit 
entirely  as  soon  as  its  speed  exceeds  the  danger  limit.  The 
relatively  safest  arrangement  is  separate  excitation  of  the 
inverted  converter  by  an  exciter  mechanically  driven  thereby, 
since  an  increase  of  speed  increases  the  exciter  voltage  at  a 
still  higher  rate,  and  thereby  the  excitation  of  the  converter, 
and  thus  tends  to  check  its  speed. 

This  danger  of  racing  does  not  exist  if  the  inverted 
converter  operates  in  parallel  with  alternating  generators, 
provided  that  the  latter  and  their  prime  movers  are  of  such 
size  that  they  cannot  be  carried  away  in  speed  by  the 
converter.  In  an  inverted  converter  running  in  parallel 
with  alternators,  the  speed  is  not  changed  by  the  field  exci- 
tation, but  a  change  of  the  latter  merely  changes  the  phase 
relation  of  the  alternating  current  supplied  by  the  converter. 
That  is,  the  converter  receives  power  from  the  direct  current 
system,  and  supplies  power  into  the  alternating  system,  but 
at  the  same  time  receives  wattless  current  from  the  alter- 
nating system,  lagging  at  under-excitation,  leading  at  over- 
excitation,  and  can  in  the  same  way  as  an  ordinary  converter 


248  ELECTRICAL   ENGINEERING. 

or  synchronous  motor  be  used  to  compensate  for  wattless 
currents  in  other  parts  of  the  alternating  system. 

IX.    Double  Current  Generators. 

Similar  in  appearance  to  the  converter,  which  changes 
from  alternating  to  direct  current,  and  to  the  inverted  con- 
verter, which  changes  from  direct  to  alternating  current,  is 
the  double  current  generator  ;  that  is,  a  machine  driven  by 
mechanical  power  and  producing  direct  current  as  well  as 
alternating  current  from  the  same  armature,  which  is  con- 
nected to  commutator  and  collector  rings  in  the  same  way 
as  in  the  converter.  Obviously  the  use  of  the  double  cur- 
rent generator  is  limited  to  those  sizes  and  speeds  at  which 
a  good  direct  current  generator  can  be  built  with  the  same 
number  of  poles  as  a  good  alternator  ;  that  is,  low  frequency 
machines  of  large  output  and  relatively  high  speed,  while 
high  frequency  slow  speed  double  current  generators  are 
undesirable. 

The  essential  difference  between  double  current  gen- 
erator and  converter  is,  however,  that  in  the  former  the 
direct  current  and  the  alternating  current  are  not  in  oppo- 
sition as  in  the  latter,  but  in  the  same  direction,  and  the 
resultant  armature  polarization  thus  the  sum  of  the  armature 
polarization  of  the  direct  current  and  of  the  alternating 
current. 

Since  at  the  same  output  and  the  same  field  strength, 
the  armature  polarization  of  the  direct  current  and  that  of 
the  alternating  current  is  the  same,  it  follows  that  the  resul- 
tant armature  polarization  of  the  double  current  generator  is 
proportional  to  the  load,  regardless  of  the  proportion  in  which 
this  load  is  distributed  between  the  alternating  and  direct 
current  side.  The  heating  of  the  armature  due  to  its  resist- 
ance depends  upon  the  sum  of  the  two  currents  ;  that  is, 
upon  the  total  load  on  the  machine.  Hence,  the  output  of 
the  double  current  generator  is  limited  by  the  current  heat- 


SYNCHRONOUS   CONVERTERS.  249 

ing  of  the  armature,  and  by  the  field  distortion  due  to  the 
armature  reaction,  in  the  same  way  as  in  a  direct  current 
generator  or  alternator,  and  is  consequently  much  less  than 
that  of  a  converter. 

In  double  current  generators,  owing  to  the  existence  of 
armature  reaction  and  consequent  field  distortion,  the  com- 
mutator brushes  are  more  or  less  shifted  against  the  neutral, 
and  the  direction  of  the  continuous  current  armature  polar- 
ization is  thus  shifted  against  the  neutral  by  the  same  angle 
as  the  brushes.  The  direction  of  the  alternating  current 
armature  polarization,  however,  is  shifted  against  the  neutral 
by  the  angle  of  phase  displacement  of  the  alternating  cur- 
rent. In  consequence  thereof,  the  reaction  upon  the  field 
of  the  two  parts  of  the  armature  polarization,  that  due 
to  the  continuous  current  and  that  due  to  the  alternating 
current,  are  usually  different.  The  reaction  on  the  field  of 
the  direct  current  load  can  be  overcome  by  a  series  field. 
The  reaction  on  the  field  of  the  alternating  current  load 
when  feeding  converters,  can  be  compensated  for  by  a 
change  of  phase  relation,  by  means  of  a  series  field  on  the 
converter,  with  self-induction  in  the  alternating  lines. 

Thus,  a  double  current  generator  feeding  on  the  alter- 
nating side  converters,  can  be  considered  as  a  direct  current 
generator  in  which  a  part  of  the  commutator,  with  a  cor- 
responding part  of  the  series  field,  is  separated  from  the 
generator  and  located  at  a  distance,  connected  by  alternating 
leads  to  the  generator.  Obviously,  automatic  compounding 
of  a  double  current  generator  is  feasible  only  if  the  phase 
relation  of  the  alternating  current  changes  from  lag  at  no 
load  to  lead  at  load,  in  the  same  way  as  produced  by  a  com- 
pounded converter.  Otherwise,  rheostatic  control  *  of  the 
generator  is  necessary.  This  is,  for  instance,  the  case  if  the 
voltage  of  the  double  current  generator  has  to  be  varied  to 
suit  the  conditions  of  its  direct  current  load,  and  the  voltage 
of  the  converter  at  the  end  of  the  alternating  lines  varied  to 
suit  the  conditions  of  load  at  the  receiving  end,  independent 


250  ELECTRICAL   ENGINEERING. 

of  the  voltage  at  the  double  current  generator,  by  means  of 
alternating  potential  regulators  or  compensators. 

Compared  with  the  direct  current  generator,  the  field  of 
the  double  current  generator  must  be  such  as  to  give  a  much 
greater  stability  of  voltage,  owing  to  the  strong  demagne- 
tizing effect  which  may  be  exerted  by  lagging  currents  on 
the  alternating  side,  and  may  cause  the  machine  to  lose  its 
excitation  altogether.  For  this  reason  it  is  frequently  prefer- 
able to  separately  excite  double  current  generators. 

X.    Conclusion. 

Of  the  types  of  machines,  converter,  inverted  converter,, 
and  double  current  generator,  sundry  combinations  can  be 
devised  with  each  other  and  with  synchronous  motors,  alter- 
nators, direct  current  motors  and  generators.  Thus,  for 
instance,  a  converter  can  be  used  to  supply  a  certain  amount 
of  mechanical  power  as  synchronous  motor.  Tn  this  case 
the  alternating  .current  is  increased  beyond  the  value  cor- 
responding to  the  direct  current  by  the  amount  of  cur- 
rent giving  the  mechanical  power,  and  the  armature  reactions 
do  not  neutralize  each  other,  but  the  reaction  of  the  alter- 
nating current  exceeds  that  of  the  direct  current  by  the 
amount  corresponding  to  the  mechanical  load.  In  the  same 
way  the  current  heating  of  the  armature  is  increased.  An 
inverted  converter  can  also  be  used  to  supply  some  mechani- 
cal power.  Either  arrangement,  however,  while  quite  fea- 
sible, has  the  disadvantage  of  interfering  with  automatic 
control  of  voltage  by  compounding. 

Double  current  generators  can  be  used  to  supply  more 
power  into  the  alternating  circuit  than  is  given  by  their 
prime  mover,  by  receiving  power  from  the  direct  current 
side.  In  this  case  a  part  of  the  alternating  power  is  gene- 
rated from  mechanical  power,  and  the  other  converted  from 
direct  current  power,  and  the  machine  combines  the  features 
of  an  alternator  with  that  of  an  inverted  converter.  Con- 


SYNCHRONOUS  CONVERTERS.  251 

versely,  when  supplying  direct  current  power  and  receiving 
mechanical  power  from  the  prime  mover  and  electric  power 
from  the  alternating  system,  the  double  current  generator 
combines  the  features  of  a  direct  current  generator  and  a 
converter.  In  either  case  the  armature  reaction,  etc.,  is  the 
sum  of  those  corresponding  to  the  two  types  of  machines 
combined. 

The  use  of  the  converter  to  change  from  alternating  to 
alternating  of  a  different  phase,  as,  for  instance,  when  using 
a  quarter-phase  converter  to  receive  power  by  one  pair  of 
its  collector  rings  from  a  single-phase  circuit,  and  supplying 
from  its  other  pair  of  collector  rings  the  other  phase  of  a 
quarter-phase  system,  or  a  three-phase  converter  on  a  single- 
phase  system  supplying  the  third  wire  of  a  three-phase 
system  from  its  third  collector  ring,  is  outside  the  scope 
of  this  treatise,  and  is,  moreover,  of  very  little  importance, 
since  induction  or  synchronous  motors  are  superior  in  this 
respect. 

APPENDIX. 
Direct  Current  Converter. 

If  n  equidistant  pairs  of  diametrically  opposite  points  of  a 
commutating  machine  armature  are  connected  to  the  ends  of 
n  compensators  or  auto-transformers,  that  is,  electric  circuits 
interlinked  with  a  magnet  circuit,  and  the  centers  of  these 
compensators  connected  with  each  other  to  a  neutral  point, 
as  shown  diagram m at ically  in  Fig.  119  lor  n  =  3,  this  neutral 
is  equidistant  in  potential  from  the  two  sets  of  commutator 
brushes,  and  such  a  machine  can  be  used  as  continuous  cur- 
rent converter,  to  transform  in  the  ratio  of  potentials  1  :  2  or 
2  : 1  or  1 : 1,  in  the  latter  case  transforming  power  from  one 
side  of  a  three-wire  system  to  the  other  side. 

Obviously  either  the  n  compensators  can  be  stationary 
and  connected  to  the  armature  by  2»  collector  rings,  or  the 
compensators  rotated  with  the  armature  and  their  common 


252 


ELECTRICAL   ENGINEERING. 


neutral  connected  to  the  external  circuit  by  one  collector 
ring. 

The  distribution  of  potential  and  of  current  in  such  a 
direct  current  converter  is  shown  in  Fig.  120  for  n  —  2,  that 
is,  two  compensators  in  quadrature. 


Fig.  119. 


Fig.  120. 


With  the  voltage  2e  between  the  outside  conductors  of 
the  system,  the  voltage  between  the  neutral  and  outside  con- 
ductor is  ±  e,  that  on  each  of  the  2«  compensator  sections  is, 


=  0,1,2  ...  2»-l. 


SYNCHRONOUS   CONVERTERS.  253 

Neglecting  losses  in  converter  and  compensator,  the  cur- 
rents in  the  two  sets  of  commutator  brushes  are  equal  and  of 
the  same  direction,  that  is,  both  outgoing  or  both  inflowing, 
and  opposite  to  the  current  in  the  neutral.  That  is,  two 
equal  currents  i  enter  the  commutator  brushes  and  issue  as 
current  2t  from  the  neutral,  or  inversely. 

From  the  law  of  conservation  of  energy  it  follows  that 
the  current  2z  entering  from  the  neutral  divides  in  2n 

equal    and    constant    branches    of   direct    current,  -  in  the 

n 

Zn  compensator  sections,  and  hence  enters  the  armature,  to 
issue  as  current  i  from  each  of  the  commutator  brushes. 
In  reality  the  current  in  each  compensator   section   is 


cos   <>  —  CD  —  —  +  a  ), 
n  n          / 


-  +    V2 


where  z'0  is  the  exciting  current  of  the  magnetic  circuit  of 
the  compensator,  and  a  the  angle  of  hysteretic  advance  of 
phase.  At  the  commutator  the  current  on  the  motor  side 
is  larger  than  the  current  on  the  generator  side,  by  the 
amount  required  to  cover  the  losses  of  power  in  converter 
and  compensator. 

In  Fig.  120  the  positive  side  of  the  system  is  generator, 
the  negative  side  motor.  This  machine  can  be  considered 
as  receiving  current  i  at  voltage  e  from  the  negative  side  of 
the  system,  and  transforming  it  into  current  i  at  voltage  e 
on  the  positive  side  of  the  system,  or  it  can  be  considered 
as  receiving  current  i  at  voltage  2e  from  the  system,  and 
transforming  into  current  2z  at  the  voltage  e  on  the  positive 
side  of  the  system,  or  of  receiving  current  2i  at  voltage  e 
from  the  negative  side,  and  returning  current  i  at  voltage  Ze. 
In  either  case  the  direct  current  converter  produces  a  dif- 
ference of  power  of  2ie  between  the  two  sides  of  the  three- 
wire  system. 

The  armature  reaction  of  the  currents  from  the  generator 
side  of  the  converter  is  equal  but  opposite  to  the  armature- 
reaction  of  the  corresponding  currents  entering  the  motor 


254 


ELECTRICAL   ENGINEERING. 


side,  and  the  motor  and  generator  armature  reactions  thus 
neutralize  each  other,  as  in  the  synchronous  converter,  that 
is,  the  resultant  armature  reaction  of  the  continuous  current 
converter  is  practically  zero,  or  the  only  remaining  armature 
reaction  is  that  corresponding  to  the  relatively  small  current 
required  to  rotate  the  machine,  that  is,  to  supply  the  internal 


MM! 

i  i  M  i 

M   1    1    M    M 

!    !   M   1   1   1    1   i 

1   i   1   1    1   1   1 

uuuuj 

UbJbjyyi 

iHJyWUUUlflM 

KJlaJbJUUJlsJUJyiaiU, 

uuuuuu 

\ 

C 

,  ,  nnnn 

' 

5i 

Fig.  121. 

losses  in  the  same.  Obviously  it  also  remains  the  armature 
reaction  of  the  current  supplying  the  electric  power  trans- 
formed into  mechanical  power,  if  the  machine  is  used  simul- 
taneously as  motor,  as  for  driving  a  booster  connected  into 
the  system  to  produce  a  difference  between  the  voltages  of 
the  two  sides,  or  the  armature  reaction  of  the  currents  gen- 
erated from  mechanical  power  if  the  machine  is  driven  as 
generator. 

While  the  currents  in  the  armature  coils  are  more  or  less 
sine  waves  in  the  alternator,  rectangular  reversed  currents 
in  the  direct  current  generator  or  motor,  and  distorted 


S  YNCHR  ONO  US   CONVER  TERS. 


255 


triple  frequency  currents  in  the  synchronous  converter,  the 
currents  in  the  armature  coils  of  the  direct  current  con- 
verter are  triangular  double  frequency  waves. 

Let  Fig.  121  represent  a  development  of  a  direct  cur- 
rent converter  with  brushes  Bl  and  B»  and  C  one  compen- 


Flg.  122. 

sat  or  receiving  current  2*  from  the  neutral.  Consider  first 
an  armature  coil  #x  adjacent  behind  (in  the  direction  of  rota- 
tion) a  compensator  lead  br  In  the  moment  'where  com- 
pensator leads  b^  bz  coincide  with  the  brushes  Bl  Bz  the  cur- 
rent i  directly  enters  the  brushes  and  coil  al  is  without 
current.  In  the  next  moment  (Fig.  12L4)  the  total  cur- 
rent i  from  bl  passes  coil  al  to  brush  Bl  while  practically  no 
current  yet  goes  from  ^  over  coils  a'  a",  etc.,  to  brush  B2. 
But  with  the  forward  motion  of  the  armature  less  and  less  of 
the  current  from  bl  passes  through  a^  av  etc.,  to  brush  Bl 
and  more  over  a'  a",  etc.,  to  brush  Bv  until  in  the  position 
of  rfj  midway  between  ^and  bz  (Fig.  121^),  one  half  of  the 
current  from  ^  passes  at  av  etc.,  to  Bv  the  other  half  af  a", 
etc.,  to  Bv  With  the  further  rotation  the  current  in  a^ 


256  ELECTRICAL   ENGINEERING. 

grows  less  and  becomes  zero  when  b^  coincides  with  Bv  or 
half  a  cycle  after  its  coincidence  with  Br  That  is,  the  cur- 
rent in  coil  ^  has  the  triangular  form  shown  as  /\  in  Fig.  122, 
changing  twice  per  period  from  0  to  i.  It  is  shown  negative, 
since  it  flows  against  the  direction  of  rotation  of  the  arma- 
ture. In  the  same  way  we  see  that  the  current  in  the  coil  a', 
adjacent  ahead  of  the  lead  bv  has  a  shape  shown  as  ir  in  Fig. 
16.  The  current  in  coil  a0  midway  between  two  commutator 
leads  has  the  form  z0,  and  in  general  the  current  in  any 
armature  coil  ax,  distant  by  angle  <o  from  the  midway  position 
<?0,  has  the  form  4,  Fig.  122. 

All  the  currents  become  zero  at  the  moment  where  the 
compensator  leads  b^b^  coincide  with  the  brushes  B^By  and 
change  by  i  at  the  moment  where  their  respective  coils  pass 
a  commutator  brush.  Thus  the  lines  A  and  A'  in  Fig.  -123- 


Fig.  123. 

with  zero  values  at  BJ3»  the  position  of  brushes,  represent 
the  currents  in  the  individual  armature  coils.  The  current 
changes  from  A  to  A'  at  the  moment  o>,  where  the  respec- 
tive armature  coil  passes  the  brush,  twice  per  period. 

With  n  compensators,  each  compensator  lead  carries  the 

current  -,  which  passes  through  the  armature   coils  as  tri- 

n  i 

angular  current,  changing  by  -  in  the  moment  the  armature 

coil  passes  a  commutator  brush.  This  current  passes  the 
zero  value  in  the  moment  the  compensator  lead  coincides 
with  a  brush.  Thus,  the  different  currents  of  n  compen- 
sators, which  are  superposed  in  an  armature  coil  ax,  have  the 


S  YNCHR  ONO  US   CONVER  TERS. 


257 


shape  shown  in  Fig.  124  for  n  =  3.     That  is,  each  compen- 
sator gives  a  set  of  slanting  lines  A1A\9  A^A' '2,  ABAf^  and  all 


Fig.  124. 

the  branch  currents  it  iz  iv  superposed,    give   a   resultant 
current  im  which  changes  by  i  in  the  moment  the  coil  passes 

the  brush.     ix  varies  between  the  extreme  values  —  (2/  —  1) 

i 

and     (2/+1),  if    the  armature  coil  is  displaced   from  the 

midway  position  between  two  adjacent  compensator  leads  by 

angle  <o,  and/  =    .     /  varies  between  —  —  and+— . 
Tf  Ln  2>n 

Thus  the  current  in  an  armature  coil  in  position  /  =  - 
can  be  denoted  in  the  range  from  /  to  1  +/,  or  wto  ?r  +o>, 


where 


*.*. 

7T 


258  ELECTRICAL   ENGINEERING 

The  effective  value  of  this  current  is, 


=  gVT+47- 

Since  in  the  same  machine  as  direct  current  generator 
at  voltage    2  e  and    current   t,   the    current    per  armature 

coil  is  — ,  the  ratio  of  current  is, 


2 

and  thus  the  relative  Pr  loss  or  the  heat  developed  in  the 
armature  coil, 


with  a  minimum, 

and  a  maximum, 

1 


1   .    !_ 

3T          o 
H 


The  mean  heating  or  I2r  of  the  armature  is  found  by  inte- 
grating over  y  from 

=-  t0       =  +        > 


as 


This  gives  the  following  table,  for  the  direct  current  con- 
verter, of  minimum  current  heating,  y0,  in  the  coil  mid- 
way between  adjacent  commutator  leads,  maximum  current 
heating,  ym,  in  the  coil  adjacent  to  the  commutator  lead, 


SYNCHRONOUS   CONVERTERS. 


259 


mean  current  heating,  r,  and  rating  as  based  on  mean  cur- 
rent heating  in  the  armature,  — - : 

Vr 


DIRECT    CURRENT    CONVERTER    7V    RATING. 

No.  of  compensators,  n  — 

d.  c 

1 

2 

3 

4 

11 

CO 

gen 

Minimum  current  heating, 

/  =  0,                           70  = 

1 

J 

^ 

J 

i 

i 

3 

i 

Maximum  current  heating, 

/  =  +  -  ,                     7m  = 

1 

4 
:j 

A 

t 

il 

j           1 

1 

Mean     current      healing 

r  = 

1 

i 

A 

H 

ii 

i  +  U172 

i 

Rating,                      -L  = 

1 

1.225 

1.549 

1.643 

1.C81 

/  3«2 

V* 

yi+«2 

As  seen,  the  output  of  the  direct  current  converter 
is  greater  than  that  of  the  same  machine  as  generator. 
Using  more  than  three  compensators  offers  very  little 
advantage,  and  the  difference  between  three  and  two  com- 
pensators is  comparatively  small,  also,  but  the  difference 
between  two  and  one-  compensator,  especially  regarding  the 
local  armature  heating,  is  considerable,  so  that  for  most 
practical  purposes  a  two-compensator  converter  would  be 
preferable. 

The  number  of  compensators  used  in  the  direct  cur- 
rent converter  has  a  similar  effect  regarding  current  dis- 
tribution, heating,  etc.,  as  the  number  of  phases  in  the 
synchronous  converter. 

Obviously  these  relative  outputs  given  in  above  table  re- 
fer to  the  armature  heating  only.  Regarding  commutation, 
the  total  current  at  the  brushes  is  the  same  in  the  converter 
as  in  the  generator,  the  only  advantage  of  the  former  thus 
the  better  commutation  due  to  the  absence  of  armature 
reaction. 


260  ELECTRICAL   ENGINEERING. 


;Kmit  ;of  ..output  set  by  armature  reaction  and  corre- 
sp(#iding  field  excitation  in  a  motor  or  generator  obviously 
do'3s  not  exist  at  all  in  a  converter.  It  follows  herefrom 
that  a."  direct  current  motor  or  generator  does  not  give  the 
most  advantageous  direct  current  converter,  but  that  in  the 
direct  current  converter,  just  as  in  the  synchronous  con- 
verter, it  is  preferable  to  proportion  the  parts  differently,  in 
accordance  with  above  discussion,  as,  for  instance,  to  use 
less  conductor  section,  a  greater  number  of  conductors  in 
series  per  pole,  etc. 


INDUCTION  MACHINES.  261 


D.  INDUCTION  MACHINES. 
I.  General. 

THE  direction  of  rotation  of  a  direct  current  motor, 
whether  shunt  or  series  wound,  is  independent  of  the  direc- 
tion of  the  current  supplied  thereto.  That  is,  when  revers- 
ing the  current,  in  a  direct  current  motor  the  direction  of 
rotation  remains  the  same.  Thus  theoretically  any  con- 
tinuous current  motor  should  operate  also  with  alternating 
-currents.  Obviously  in  this  case  not  only  the  armature,  but 
also  the  magnetic  field  of  the  motor  must  be  thoroughly 
laminated  to  exclude  eddy  currents,  and  care  taken,  that  the 
.current  in  field  and  in  armature  reverses  simultaneously. 
The  simplest  way  of  fulfilling  the  latter  condition  is  obviously 
to  connect  field  and  armature  in  series  as  alternating  current 
series  motor.  Such  motors  have  been  used  to  a  limited  ex- 
tent. Their  main  objection  is,  however,  the  excessive  self- 
induction  introduced  by  the  alternating  field  excitation,  and 
the  consequently  low  power  factor,  and  also  the  vicious 
sparking  at  the  commutator.  Besides,  as  series  motor  the 
speed  is  not  constant,  but  depends  upon  the  load. 

The  shunt  motor  on  an  alternating  circuit  has  the  ob- 
jection that  in  the  armature  the  current  should  be  energy 
current,  thus  in  phase  with  the  E.M.F.,  while  in  the  field 
the  current  is  lagging  nearly  90°,  as  magnetizing  current. 
Thus  field  and  armature  would  be  out  of  phase  with  each 
other.  To  overcome  this  objection  the  field  may  be  excited 
from  a  separate  E.M.F.  differing  90°  in  phase  from  that 
supplied  to  the  armature.  That  is,  the  armature  of  the 
motor  is  fed  by  one,  the  field  by  the  other  phase  of  a  quarter- 


262  ELECTRICAL   ENGINEERING. 

phase  system,  and  thus  the  current  in  the  armature  brought 
approximately  in  phase  with  the  magnetic  flux  of  the  field. 

Such  an  arrangement  obviously  loads  the  two  phases  of 
the  system  unsymmetrically,  the  one  with  the  armature 
energy  current,  the  other  with  the  lagging  field  current.  To 
balance  the  system  two  such  motors  may  be  used  simultane- 
ously and  combined  in  one  structure,  the  one  receiving 
energy  current  from  the  first,  magnetizing  current  from  the 
second  phase,  the  second  motor  receiving  magnetizing  cur- 
rent from  the  first  and  energy  current  from  the  second 
phase. 

The  objection  of  the  vicious  sparking  of  the  commutator 
can  be  entirely  overcome  by  utilizing  the  alternating  feature 
of  the  current ;  that  is,  instead  of  leading  the  current  into 
the  armature  by  commutator  and  brushes,  producing  it 
therein  by  electro-magnetic  induction,  by  closing  the  arma- 
ture conductors  upon  themselves  and  surrounding  the 
armature  by  an  inducing  coil  at  right  angles  to  the  field 
exciting  coil. 

Such  motors  have  been  built,  consisting  of  two  struc- 
tures each  containing  a  magnetizing  circuit  acted  upon  by 
one  phase,  and  an  energy  circuit  inducing  a  closed  circuit 
armature  and  excited  by  the  other  phase  of  a  quarter-phase 
system. 

Going  still  a  step  further,  the  two  structures  can  be  com- 
bined into  one  by  having  each  of  the  two  ceils  fulfill  the 
double  function  of  magnetizing  the  f.eld  and  inducing  cur- 
rents in  the  armature  which  are  acted  upon  by  the  mag- 
netization produced  by  the  other  phase. 

Obviously,  instead  of  two  phases  in  quadrature  any 
number  of  phases  can  be  used. 

This  leads  us  by  gradual  steps  of  development  from  the 
continuous  current  shunt  motor  to  the  alternating  current 
polyphase  induction  motor. 

In  its  general  behavior  the  alternating  current  induction 
motor  is  analogous  to  the  continuous  current  shunt  motor. 


INDUCTION  MACHINES.  263 

Like  the  shunt  motor,  it  operates  at  approximately  constant 
magnetic  density.  It  will  run  at  fairly  constant  speed,  slow- 
ing down  gradually  with  increasing  load.  The  main  differ- 
ence is,  that  in  the  induction  motor  the  current  is  not  passed 
into  the  armature  by  a  system  of  brushes,  as  in  the  con- 
tinuous current  motor,  but  induced  in  the  armature  ;  and  in 
consequence  thereof,  the  primary  circuit  of  the  induction 
motor  fulfills  the  double  function  of  an  exciting  circuit 
corresponding  to  the  field  circuit  of  the  continuous  current 
machine,  and  a  primary  circuit  inducing  a  secondary  current 
in  the  armature  by  electro-magnetic  induction. 

Since  in  the  armature  of  the  induction  motor  the  cur- 
rents are  produced  by  induction  from  the  primary  impressed 
currents,  the  induction  motor  in  its  electro-magnetic  fea- 
tures is  essentially  a  transformer;  that  is,  it  consists  of  a 
magnetic  circuit  or  magnetic  circuits  interlinked  with  two 
electric  circuits  or  sets  of  circuits,  the  primary  or  inducing, 
and  the  secondary  or  induced  circuit.  The  difference  be- 
tween transformer  and  induction  motor  is,  that  in  the  former 
the  secondary  is  fixed  regarding  the  primary,  and  the  electric 
energy  induced  in  the  secondary  is  made  use  of,  while  in  the 
latter  the  secondary  is  movable  regarding  the  primary,  and 
the  mechanical  force  acting  between  primary  and  secondary 
is  used.  In  consequence  thereof  the  frequency  of  the  cur- 
rents flowing  in  the  secondary  of  the  induction  motor  differs 
from,  and  as  a  rule  is  very  much  lower  than,  that  of  the 
currents  impressed  upon  the  primary,  and  thus  the  ratio  of 
E.M.F.'s  induced  in  primary  and  in  secondary  is  not  the 
ratio  of  their  respective  turns,  but  is  the  ratio  of  the  product 
of  turns  and  frequency. 

Taking  due  consideration  of  this  difference  of  frequency 
between  primary  and  secondary,  the  theoretical  investigation 
of  the  induction  motor  corresponds  to  that  of  the  stationary 
transformer.  The  transformer  feature  of  the  induction 
motor  predominates  to  such  an  extent  that  in  theoretical 
investigation  the  induction  motor  is  best  treated  as  a  trans- 


264  ELECTRICAL   ENGINEERING. 

former,  and  the  electrical  output  of  the  transformer  corre- 
sponds to  the  mechanical  output  of  the  induction  motor. 

The  secondary  or  armature  of  the  motor  consists  of  two 
or  more  circuits  displaced  in  phase  from  each  other  so  as  to 
offer  a  closed  secondary  to  the  primary  circuits,  irrespective 
of  the  relative  motion.  The  primary  consists  of  one  or 
several  circuits. 

In  consequence  of  the  relative  motion  of  the  primary  and 
secondary,  the  magnetic  circuit  of  the  induction  motor  must 
be  arranged  so  that  the  secondary  while  revolving  does  not 
leave  the  magnetic  field  of  force.  That  means,  the  magnetic 
field  of  force  must  be  of  constant  intensity  in  all  directions, 
or,  in  other  words,  the  component  of  magnetic  flux  in  any 
direction  in  space  be  of  the  same  or  approximately  the  same 
intensity  but  differing  in  phase.  Such  a  magnetic  field  can 
either  be  considered  as  the  superposition  of  two  magnetic 
fields  of  equal  intensity  in  quadrature  in  time  and  space,  or 
it  can  be  represented  theoretically  by  a  revolving  magnetic 
flux  of  constant  intensity,  or  simply  treated  as  alternating 
magnetic  flux  of  the  same  intensity  in  every  direction. 

In  the  polyphase  induction  motor  this  magnetic  field  is 
produced  by  a  number  of  electric  circuits  relatively  displaced 
in  space,  and  excited  by  currents  having  the  same  displace- 
ment in  phase  as  the  exciting  coils  have  in  space. 

In  the  monocyclic  motor  one  of  the  two  superimposed 
quadrature  fields  is  excited  by  the  primary  energy  circuit, 
the  other  by  the  magnetizing  or  teaser  circuit. 

In  the  single-phase  motor  one  of  the  two  superimposed 
magnetic  quadrature  fields  is  excited  by  the  primary  elec- 
tric circuit,  the  ether  by  the  induced  secondary  or  armature 
currents  carried  into  quadrature  position  by  the  rotation  of 
the  secondary.  In  either  case,  at  or  near  synchronism  the 
magnetic  fields  are  identical. 

The  transformer  feature  being  predominant,  in  theo- 
retical investigations  of  induction  motors  it  is  generally 
preferable  to  start  therefrom. 


INDUCTION  MACHINES.  265 

The  characteristics  of  the  transformer  are  independent 
of  the  ratio  of  transformation,  other  things  being  equal. 
That  is,  doubling  the  number  of  turns  for  instance,  and  at 
the  same  time  reducing  their  cross-section  to  one-half,  leaves 
the  efficiency,  regulation,  etc.,  of  the  transformer  unchanged. 
In  the  same  way,  in  the  induction  motor  it  is  unessential 
what  the  ratio  of  primary  to  secondary  turns  is,  or  in  other 
words,  the  secondary  circuit  can  be  wound  for  any  suitable 
number  of  turns,  provided  the  same  total  copper  cross-section 
is  used.  In  consequence  hereof  the  secondary  circuit  is 
mostly  wound  with  one  or  two  bars  per  slot,  to  get  maxi- 
mum amount  of  copper,  that  is,  minimum  resistance  of 
secondary. 

The  general  characteristics  of  the  induction  motor  being 
independent  of  the  ratio  of  turns,  it  is  for  theoretical  con- 
siderations simpler  to  assume  the  secondary  motor  circuits 
reduced  to  the  same  number  of  turns  and  phases  as  the  pri- 
mary,  or  the  ratio  of  transformation  1,  by  multiplying 
all  secondary  currents  and  dividing  all  E.M.F.'s  with  the 
ratio  of  turns,  multiplying  all  secondary  impedances  by 
the  square  of  the  ratio  of  turns,  etc. 

Thus  in  the  following  under  secondary  current,  E.M.F., 
impedance,  etc.,  shall  always  be  understood  their  values 
reduced  to  the  primary,  or  corresponding  to  a  ratio  of  turns  1 
to  1,  and  the  same  number  of  secondary  as  primary  phases, 
although  in  practice  a  ratio  1  to  1  will  hardly  ever  be  used, 
as  not  fulfilling  the  condition  of  uniform  magnetic  reluctance 
desirable  in  the  starting  of  the  induction  motor. 


II.  Polyphase  Induction  Motor. 
1.   INTRODUCTION. 

The  typical  induction  motor  is  the  polyphase  motor.  By 
gradual  development  from  the  direct  current  shunt  motor  we 
arrive  at  the  polyphase  induction  motcr. 


266  ELECTRICAL   ENGINEERING. 

The  magnetic  field  of  any  induction  motor,  whether  sup- 
plied by  polyphase,  monocyclic,  or  single-phase  E.M.F.,  is  at 
normal  condition  of  operation,  that  is,  near  synchronism,  a 
polyphase  field.  Thus  to  a  certain  extent  all  induction 
motors  can  be  called  polyphase  machines.  When  supplied 
with  a  polyphase  system  of  E.M.F.'s  the  internal  reactions  of 
the  induction  motor  are  simplest  and  only  those  of  a  trans- 
former with  moving  secondary,  while  in  the  single-phase 
induction  motor  at  the  same  time  a  phase  transformation 
occurs,  the  second  or  magnetizing  phase  being  produced 
from  the  impressed  phase  of  E.M.F.  by  the  rotation  of  the 
motor,  which  carries  the  induced  currents  in  quadrature 
position  with  the  inducting  current. 

The  polyphase  induction  motor  of  the  three-phase  or 
quarter-phase  type  is  the  one  most  commonly  used,  while 
single-phase  motors  have  found  a  more  limited  application 
only,  and  especially  for  smaller  powers. 

Thus  in  the  following  more  particularly  the  polyphase 
induction  machine  shall  be  treated,  and  the  single-phase  type 
only  in  so  far  discussed  as  it  differs  from  the  typical  poly- 
phase machine. 

2.  CALCULATION. 

In  the  polyphase  induction  motor, 
Let 

YQ  =  g  -f-  jb  =  primary   exciting   admittance,   or   admit- 
tance  of  the   primary   circuit  at  open   secondary 
circuit ; 
that  is, 

ge  =  magnetic  energy  current, 
be  =  wattless  magnetizing  current, 
where 

e  =  induced  E.M.F.  of  motor, 
ZQ  =  r0  —  jxQ  =  primary  self-inductive  impedance, 
Zl  =  r^  —  jxi  =  secondary  self-inductive  impedance, 

reduced  to  the  primary  by  the  ratio  of  turns.* 

*  The  self-inductive  reactance  refers  to  that  flux  which  surrounds  one  of  the  electric 
circuits  only,  without  being  interlinked  with  the  other  circuits. 


INDUCTION  MACHINES.  267 

All  these  quantities  refer  to  one  primary  circuit  and  one 
corresponding  secondary  circuit.  Thus  in  a  three-phase  in- 
duction motor  the  total  power,  etc.,  is  three  times  that  of 
one  circuit,  in  the  quarter-phase  motor  with  three-phase 
armature  1^  of  the  three  secondary  circuits  are  to  be 
considered  as  corresponding  to  each  of  the  two  primary 
circuits,  etc. 

Let 

e  =  primary  counter  E.M.F.,  or  E.M.F.  induced  in  the  pri- 
mary circuit  by  the  flux  interlinked  with  primary  and 
secondary  (mutual  induction)  ; 

s  =  slip,  with  the  primary  frequency  as  unit, 

that  is, 

s  =  0,  denoting  synchronous  rotation, 
s  =  1  standstill  of  the  motor. 

We  then  have, 

1  —  s  =  speed  of   the  motor  secondary  as  fraction  of   syn- 

chronous speed. 

sW  =  frequency  of  secondary  currents, 
where 

JV  =  frequency  impressed  upon  the  primary  ; 
hence, 

se  =  E.M.F.  induced  in  the  secondary. 

The  actual  impedance  of  the   secondary  circuit  at  the 

frequency  sN  is, 

Zf  =  r^ 
hence, 

Secondary  current, 

T       se  se  I       si\  ••'.-.    Px^    \ 

£=-=-=-    —.  —  =  e      0,00  +/    2   ,     o    2  1  =*  *  0%  +J0a), 
^i       ri—jsxi         vr+^i          rf+£x£{ 

where,  srt  s2x^<   . 


Primary  exciting  current, 


268  ELECTRICAL   ENGINEERING. 

hence, 

Total  primary  current, 

/o  =  '  [(*i + *)  +/(*2  +  *)]  =*  to 
where, 


The  E.M.F.  consumed  in  the  primary  circuit  by  the 
impedance  ZQ  is  fQ  Z^  the  counter  E.M.F.  is  ^,  hence, 

Primary  terminal  voltage, 

EQ  =  e  +  70Z0  =  <?[!  +  (^  +  /&j)  (V0  —7*0)]  =  *  (<a 
where 

Eliminating  complex  quantities,  we  have, 

hence, 

\ 
Counter  E.M.F.  of  motor, 

e  =• 

where, 

-E0  =  impressed  E.M.F.,  absolute  value. 

Substituting  this  value  in  the  equations  of  fv  /oo,  fo> 
etc.,  gives  the  complex  expressions  of  currents  and  E.M.F.'s, 
and  eliminating  the  imaginary  quantities,  we  have, 

Primary  current, 

The  torque  of  the  polyphase  induction  motor  (or  any 
other  motor  or  generator)  is  proportional  to  the  product  of 
the  mutual  magnetic  flux  and  the  component  of  ampere 
turns  of  the  armature  or  secondary,  which  is  in  phase  with 
the  magnetic  flux  in  time,  but  in  quadrature  therewith  in 
direction  or  space.  Since  the  induced  E.M.F.  is  propor- 
tional to  the  mutual  magnetic  flux  and  the  number  of  turns, 
but  in  quadrature  thereto  in  time,  the  torque  of  the  induc- 
tion motor  is  proportional  also  to  the  product  of  the  induced 


INDUCTION  MACHINES.  269 

E.M.F,  and  the  component  of  secondary  current  in  quadra- 
ture therewith,  in  time  and  in  space. 
Since 


is  the  secondary  current  corresponding  to.the  induced  E.M.F. 
e,  the  secondary  current  in  the  quadrature  position  thereto 
in  space,  that  is,  corresponding  to  the  induced  E.M.F.  je,  is 


and  ave  is  ti^e  component  of  this  current  in  quadrature  in 
time  with  me  E.M.F.  e. 

Thus  the  torque  is  proportional  to  e  X  a^e,  or 

- 


^  r*  +  fx  +       (ri 

This"  value  T  is  in  its  dimension  a  power,  and  it  is  the 
power  which  the  torque  of  the  motor  would  develop  at 
synchronous  speed. 

In  induction  motors,  and  in  general  motors  which  have 
a  definite  limiting  speed,  it  is  preferable  to  give  the  torque 
in  the  form  of  the  power  developed  at  the  limiting  speed, 
in  this  case  synchronism,  as  "synchronous  watts,"  since 
thereby  it  is  made  independent  of  the  individual  conditions 
of  the  motor,  as  its  number  of  poles,  frequency,  etc.,  and 
made  comparable  with  the  power  input,  etc.  It  is  obvious 
that  when  given  in  synchronous  watts,  the  maximum  pos- 
sible value  of  torque  which  could  be  reached,  if  there  were  no 
losses  in  the  motor,  equals  the  power  input.  Thus,  in  an 
induction  motor  with  9000  watts  power  input,  a  torque  of 
7000  synchronous  watts  means  that  £  of  the  maximum  theo- 
retically possible  torque  is  realized,  while  the  statement,  "a 
torque  of  30  Ibs.  at  one  foot  radius,"  would  be  meaningless 
without  knowing  the  number  of  poles  and  the  frequency. 
Thus^the  denotation  of  the  torque  in  synchronous  watts  is 
the  most  general,  and  preferably  used  in  induction  motors. 


270  ELECTRICAL   ENGINEERING. 

Since  the  theoretically  maximum  possible  torque  equals 
the  power  input,  the  ratio, 

torque  in  synchronous  watts  output 

power  input 
that  is, 

actual  torque 
maximum  possible  torque 

is  called  the  torque  efficiency  of  the  motor,  analogous  to  the 
power  efficiency  or 

power  output  ^ 
power  input 
that  is, 

power  output 


maximum  possible  power  output 
Analogously 

torque  in  synchronous  watts 
voltamperes  input 

is  called  the  apparent  torque  efficiency. 

The  definition  of  these  quantities,  which  are  of  importance 
in  judging  induction  motors,  are  thus : 

The  "  efficiency  "  or  "power  efficiency  "  is  the  ratio  of  the 
true  mechanical  output  of  the  motor  to  the  output  which  it 
would  give  at  the  same  power  input  if  there  were  no  internal 
losses  in  the  motor. 

The  "apparent  efficiency"  or  "  apparent  power  efficiency" 
is  the  ratio  of  the  mechanical  output  of  the  motor  to  the 
output  which  it  would  give  at  the  same  voltampere  input  if 
there  were  neither  internal  losses  nor  phase  displacement  in 
the  motor. 

The  "  torque  efficiency  "  is  the  ratio  of  the  torque  of  the 
motor  to  the  torque  which  it  would  have  at  the  same  power 
input  if  there  were  no  internal  losses  in  the  motor. 

The  "apparent  torque  efficiency"  is  the  ratio  of  the 
torque  of  the  motor  to  the  torque  which  it  would  give  at  the 
same  voltampere  input  if  there  were  neither  internal  losses 
nor  phase  displacement  in  the  motor. 


INDUCTION  MACHINES.  271 

The  torque  efficiencies  are  of  special  interest  in  starting 
where  the  power  efficiencies  are  necessarily  zero,  but  it 
nevertheless  is  of  importance  to  find  how  much  torque  per 
watt  or  per  volt-ampere  input  is  given  by  the  motor. 

Since 

T=ezal 

is  the  power  developed  by  the  motor  torque  at  synchronism, 
the  power  developed  at  the  speed  of  (1  —  s)  X  synchronism, 
or  the  actual  oower  output  of  the  motor,  is, 

P  =  (1  -  s)  T 


r*  +  s-x* 

The  output  P  includes  friction,  windage,  etc.  ;  thus,  the 
net  mechanical  output  is  P  —  friction,  etc.  Since,  however, 
friction,  etc.,  depend  upon  the  mechanical  construction  of 
the  individual  motor  and  its  use,  it  cannot  be  included  in  a 
general  formula.  P  is  thus  the  mechanical  output,  and  T 
the  torque  developed  at  the  armature  conductors. 

The  primary  current 


has  the  quadrature  components  eb±  and  eby 
The  primary  impressed  E.M.F. 


has  the  quadrature  components  ec^  and  ecv 

Since  the  components  eb^  and  ccv  and  eb2  and  ecv  respec- 
tively, are  in  quadrature  with  each  other,  and  thus  represent 
no  power,  the  power  input  of  the  primary  circuit  is, 

eb±  X  ec^  +  ebz  X  ecz. 
Or, 


The  volt-amperes  or  apparent  input  is  obviously, 
-h  £22)  (cf  +  ^2). 


272  ELECTRICAL   ENGINEERING. 

Since  the  counter  E.M.F.  e  (and  thus  the  impressed 
E.M.F.  EQ)  enters  in  the  equation  of  current,  magnetism,  etc., 
as  a  simple  factor,  in  the  equations  of  torque,  power  input 
and  output  and  voltampere  input  as  square,  and  cancels  in 
the  equation  of  efficiency,  power  factor,  etc.,  it  follows  that 
the  current,  magnetic  flux,  etc.,  of  an  induction  motor  are 
proportional  to  the  impressed  E.M.F.,  the  torque,  power  out- 
put, power  input,  and  voltampere  input  are  proportional  to 
the  square  of  the  impressed  E.M.F.,  and  the  torque  and 
power  efficiencies  and  the  power  factor  are  independent  of 
the  impressed  voltage. 

In  reality,  however,  a  slight  decrease  of  efficiency  and 
power  factor  occurs  at  higher  impressed  voltages,  due  to  the 
increase  of  resistance  caused  by  the  increasing  temperature 
of  the  motor  and  due  to  the  approach  to  magnetic  saturation, 
and  a  slight  decrease  of  efficiency  occurs  at  lower  voltages 
when  including  in  the  efficiency  the  loss  of  power  by  friction 
(since  this  is  independent  of  the  output  and  thus  at  lower 
voltage,  that  is,  lesser  output,  a  larger  percentage  of  the 
output),  so  that  the  efficiencies  and  the  power  factor  can  be 
considered  as  independent  of  the  impressed  voltage,  and  the 
torque  and  power  proportional  to  the  square  thereof  only 
approximately,  but  sufficiently  close  for  most  purposes. 

3.    LOAD  AND  SPEED  CURVES. 

Diagrammatically  it  is  most  instructive  in  judging  about 
an  induction  motor  to  plot  from  the  preceding  calculation. 

1st.  The  load  cmves,  that  is,  with  the  load  or  power  out- 
put as  abscissae,  the  values  of  speed  (as  fraction  of  synchro- 
nism), of  current  input,  power  factor,  efficiency,  apparent 
efficiency,  and  torque. 

2d.  The  speed  curves,  that  is,  with  the  speed,  as  fraction 
of  synchronism,  as  abscissae,  the  values  of  torque,  current 
input,  power  factor,  torque  efficiency,  and  apparent  torque 
efficiency. 


INDUCTION  MACHINES. 


273 


The  load  curves  are  most  instructive  for  the  range  of 
speed  near  synchronism,  that  is,  the  normal  operating  con- 
ditions of  the  motor,  while  the  speed  curves  characterize  the 
behavior  of  the  motor  at  any  speed. 


INDUCTION  MOTOR 
LOAD  CURVES- 


Y0=.01  + 


110 


^ 


50 


2OOQ 


OUTPt^T 


2000     3.000     4000      5000     6OOO    7OOO 
Fig.  125. 

In  Fig.  125  are  plotted  the  load  curves,  and  in  Fig.  126 
the  speed  curves  of  a  typical  polyphase  induction  motor  of 
moderate  size,  of  the  constants, 


K0  =  .01  -  .!/, 


274 


ELECTRICAL   ENGINEERING. 


As  sample  of  a  poor  motor  of  high  resistance  and  high 
admittance  or  exciting  current  are  plotted  in  Fig.  127  the 
load  curves  of  a  motor  of  the  constants, 

e0=  110, 
K0  =  .04  +  .4/, 
Z^.3-,3/, 
Z0=.3-.3/, 

showing  the  overturn  of  the  power  factor  curve  frequently 
met  in  poor  motors. 


INDUOTIO^I  M]OTOR 

SPEED 
Z0=fZ,4.1-f-3j 


t-r- 

Q.      O  <Q 

*-  Z-r 


I- 


Fig.  126. 

The  shape  of  the  characteristc  motor  curves  depends 
entirely  on  the  three  complex  constants,  F0,  Zv  and  Z0,  but 
is  essentially  independent  of  the  impressed  voltage. 

Thus  a  change  of  the  admittance  F0  has  no  effect  on  the 
characteristic  curves  provided  that  the  impedances  Z^  and  ZQ 
are  changed  inversely  proportional  thereto,  such  a  change 
merely  representing  the  effect  of  a  change  of  impressed 
voltage.  A  change  of  one  of  the  impedances  has  relatively 
little  effect  on  the  motor  characteristics,  provided  that  the 
other  impedance  changes  so  that  the  sum  Z±  +  Z^  remains 
constant,  and  thus  the  motor  can  be  characterized  by  its 
total  internal  impedance,  that  is, 


z0, 


or, 


•r  -jx 


INDUCTION  MACHINES. 


275 


Thus  the  characteristic  behavior  of  the  induction  motor 
depends  upon  two  complex  imaginary  constants  Y  and  Z,  or 
four  real  constants,  g,  b,  r,  x,  the  same  terms  which  char- 
acterize the  stationary  alternating  current  transformer  on 
non-inductive  load. 


Fig.  127. 

Instead  of  conductance  g,  susceptance  by  resistance  ry 
and  reactance  x,  as  characteristic  constants  may  be  chosen, 

the  absolute  admittance  y  =  \lgz  -f-  ^ ; 

the  absolute  impedance   z  =  Vr2  -f  op ; 

the  power  factor  of  admittance  ft  =  g/y ; 
and  the  power  factor  of  impedance  y  =r/z-, 


276  ELECTRICAL   ENGINEERING. 

If  the  admittance  y  is  reduced  w-fold  and  the  impedance 
z  increased  «-fold,  with  the  E.M.F.  \lnEQ  impressed  upon 
the  motor,  the  speed,  torque,  power  input  and  output,  volt- 
ampere  input  and  excitation,  power  factor,  efficiencies,  etc., 
of  the  motor,  that  is,  all  its  characteristic  features  remain  the 
same,  as  seen  from  above  given  equations,  and  since  a  change 
of  impressed  E.M.F.  does  not  change  the  characteristics,  it 
follows  that  a  change  of  admittance  and  of  impedance  does 
not  change  the  characteristics  of  the  motor,  provided  the 
product  0  =  ys  remains  the  same. 

Thus  the  induction  motor  is  characterized  by  three 
constants  only  : 

The  product  of  admittance  and  impedance  0  =  yz>  which 
may  be  called  the  characteristic  constant  of  the  motor. 

The  power  factor  of  primary  admittance  /?=<£ 

r  y 

The  power  factor  of  impedance  y  =  -• 

z 

All  these  three  quantities  are  absolute  numbers. 
The  physical  meaning  of  the  characteristic  constant  or 
the  product  of  admittance  and  impedance  is  the  following  : 

If, 

^x)  =  exciting  current, 
Sw=  starting  current, 

We  have  approximately, 


~T' 

Ao 

i  _>•-&. 

Ao 

The  characteristic  constant  of  the  induction  motor  0  — 
yz  is  the  ratio  of  exciting  current  to  starting  current  or 
current  at  standstill. 

At  given  impressed  E.M.F.,  the  exciting  current  Too  is 
inversely  proportional  to  the  mutual  inductance  of  primary 
and  secondary  circuit.  The  starting  current  710  is  inversely 


INDUCTION  MACHINES.  277 

proportional  to  the  sum  of  the  self-inductance  of  primary  and 
secondary  circuit. 

Thus  the  characteristic  constant  6  =  yz  is  approximately 
the  ratio  of  total  self-inductance  to  mutual  inductance  of 
the  motor  circuits  ;  that  is,  the  ratio  of  the  flux  interlinked 
with  one  circuit,  primary  or  secondary,  only,  to  the  flux 
interlinked  with  both  circuits,  primary  and  secondary,  or 
the  ratio  of  the  waste  flux  to  the  useful  flux.  The  impor- 
tance of  this  quantity  is  evident. 

4.    EFFECT  OF  ARMATURE  RESISTANCE  AND  STARTING. 

The  secondary  or  armature  resistance  i\  enters  the  equa- 
tion of  secondary  current, 

se  I      sr,  s2xl 


and  the  further  equations  only  indirectly  in  so  far  as  r^  is 
contained  in  a^  and  #2. 

Increasing  the  armature  resistance  //-fold,  to  nrv  we  get 
at  an  ;/-fold  increased  slip  ns, 

nse  se 

nr^  —  jnsxl       r±  —  jsx  ' 

that  is,  the  same  value,  and  thus  the  same  values  for  e,  1^ 
T,  PQ,  Qy  while  the  power  is  decreased  from  P  =  (1  —  s)  T, 
to  />=(!—  ns)  T,  and  the  efficiency  and  apparent  effi- 
ciency are  correspondingly  reduced.  The  power  factor  is  not 
changed.  Hence, 

An  increase  of  armature  resistance  r±  produces  a  pro- 
portional increase  of  slip  s,  and  thereby  corresponding 
decrease  of  power  output,  efficiency  and  apparent  efficiency, 
but  does  not  change  the  torque,  power  input,  current,  power 
factor,  and  the  torque  efficiencies. 

Thus  the  insertion  of  resistance  in  the  armature  or  sec- 
ondary of  the  induction  motor  offers  a  means  to  reduce  the 
speed  corresponding  to  a  given  torque,  and  thereby  any 
desired  torque  can  be  produced  at  any  speed  below  that 


278 


ELECTRICAL   ENGINEERING. 


corresponding  to  short  circuited  armature  or  secondary,  with- 
out changing  torque  or  current. 

Hence,  given  the  speed  curve  of  a  short  circuited  motor, 
the  speed  curve  with  resistance  inserted  in  the  armature 
can  be  derived  therefrom  directly  by  increasing  the  slip  in 
proportion  to  the  increased  resistance. 

This  is  done  in  Fig.  128,  in  which  are  shown  the  speed 


128. 


INDUCTION  MACHINES.  279 

« 

curves  of  the  motor  Figs.  125  and  126,  between  standstill 
and  synchronism,  for : 

Short  circuited  armature,  1\  =  .1  (same  as  Fig.  126). 

.15  ohms  additional  resistance  per  circuit  inserted  in 
armature,  j\  =  .25,  that  is,  2.5  times  increased  slip. 

.5  ohms  additional  resistance  inserted  in  the  armature, 
^  =  .6,  that  is,  6  times  increased  slip. 

1.5  ohms  additional  resistance  inserted  in  the  armature, 
r^  =  1.6,  that  is,  16  times  increased  slip. 

The  corresponding  current  curves  are  shown  on  the 
same  sheet. 

With  short  circuited  armature  the  maximum  torque  of 
8250  synchronous  watts  is  reached  at  16%  slip.  The 
starting  torque  is  2950  synchronous  watts,  and  the  starting 
current  176  amperes. 

With  armature  resistance  1\  =  .25,  the  same  maximum 
torque  is  reached  at  40%  slip,  the  starting  torque  is  in- 
creased to  6050  synchronous  watts,  and  the  starting  current 
decreased  to  160  amperes. 

With  the  armature  resistance  1\  =  .6,  the  maximum 
torque  of  8250  synchronous  watts  takes  place  in  starting, 
and  the  starting  current  is  decreased  to  124  amperes. 

With  armature  resistance  j\  =  1.6,  the  starting  torque  is 
below  the  maximum,  5620  synchronous  watts,  and  the  start- 
ing current  is  only  64  amperes. 

In  the  two  latter  cases  the  lower  or  unstable  branch  of 
the  torque  curve  has  altogether  disappeared,  and  the  motor 
speed  is  stable  over  the  whole  range  ;  the  motor  starts  with 
the  maximum  torque  which  it  can  reach,  and  with  increas- 
ing speed,  torque  and  current  decrease ;  that  is,  the  motor 
has  the  characteristic  of  the  direct  current  series  motor, 
except  that  its  maximum  speed  is  limited  by  synchronism. 

It  follows  herefrom  that  high  secondary  resistance,  while 
very  objectionable  in  running  near  synchronism,  is  advan- 
tageous in  starting  or  running  at  very  low  speed,  by  redu- 
cing the  current  input  and  increasing  the  torque. 


280  ELECTRICAL   ENGINEERING. 

In  starting  we  have, 

*  —  1, 

Substituting  this  value  in  the  equations  of  sub-section  2  gives 
the  starting  torque,  starting  current,  etc.,  of  the  polyphase 
induction  motor. 

In  Fig.  129  are  shown  for  the  motor  in  Figs.  125,  126, 
and  128  the  values  of  starting  torque,  current,  power  fac- 


RE6IS   ANCE  ,  R 
5      1.0       16     20     2.5      30    35     40     45    5 


Fig.  129. 


INDUCTION  MACHINES.  281 

t 

tor,  torque  efficiency,  and  apparent  torque  efficiency  for 
various  values  of  the  secondary  motor  resistance,  from 
i\  =  .1,  the  internal  resistance  of  the  motor,  or  R  =  0 
additional  resistance  to  i\  =  5.1  or  R  =  5  ohms  additional 
resistance.  The  best  values  of  torque  efficiency  are  found 
beyond  the  maximum  torque  point. 

The  same  Fig.  129  also  shows  the  torque  with  resistance 
inserted  into  the  primary  circuit. 

The  insertion  of  reactance,  either  in  the  primary  or  in  the 
secondary,  is  just  as  unsatisfactory  as  the  insertion  of  resist- 
ance in  the  primary  circuit. 

Capacity  inserted  in  the  secondary  very  greatly  increases 
the  torque  within  the  narrow  range  of  capacity  correspond- 
ing to  resonance  with  the  internal  reactance  of  the  motor, 
and  the  torque  which  can  be  produced  in  this  way  is  far  in 
excess  of  the  maximum  torque  of  the  motor  when  running 
or  when  starting  with  resistance  in  the  secondary. 

But  even  at  its  best  value,  the  torque  efficiency  available 
with  capacity  in  the  secondary  is  far  below  that  available 
with  resistance. 

For  further  discussion  of  the  polyphase  induction  motor, 
see  Transactions  American  Institute  of  Electrical  Engineers, 
1897,  page"  134 

III.  Single-phase  Induction  Motor. 
1.    INTRODUCTION. 

In  the  polyphase  motor  a  number  of  secondary  coils,  dis- 
placed in  position  from  each  other,  are  acted  upon  by  a 
number  of  primary  coils  displaced  in  position  and  excited  by 
E.M.F.'s  displaced  in  phase  from  each  other  by  the  same 
angle  as  the  displacement  of  position  of  the  coils. 

In  the  single-phase  induction  motor  a  system  of  armature 
circuits  is  acted  upon  by  one  primary  coil  (or  system  of 
primary  coils  connected  in  series  or  in  parallel)  excited  by  a 
single  alternating  current. 


282  ELECTRICAL   ENGINEERING. 

A  number  of  secondary  circuits  displaced  in  position 
must  be  used  so  as  to  offer  to  the  primary  circuit  a  short 
circuited  secondary  in  any  position  of  the  armature.  If  only 
one  secondary  coil  is  used,  the  motor  is  a  synchronous 
induction  motor,  and  belongs  to  the  class  of  reaction 
machines. 

A  single-phase  induction  motor  will  not  start  from  rest, 
but  when  started  in  either  direction  will  accelerate  with 
increasing  torque  and  approach  synchronism. 

When  running  at  or  very  near  synchronism,  the  magnetic 
field  of  the  single-phase  induction  motor  is  identical  with 
that  of  a  polyphase  motor,  that  is,  can  be  represented  by  the 
theory  of  the  rotating  field.  Thus,  in  a  turn  wound  under 
angle  </>  to  the  primary  winding  of  the  single-phase  induction 
motor,  at  synchronism  an  E.M.F.  is  induced  equal  to  that 
induced  in  a  turn  of  the  primary  winding,  but  differing 
therefrom  by  angle  <j>  in  phase. 

In  a  polyphase  motor  the  magnetic  flux  in  any  direction 
is  due  to  the  resultant  M.M.F.  of  primary  and  of  secondary 
currents,  in  the  same  way  as  in  a  transformer.  The  same 
is  the  case  in  the  direction  of  the  axis  of  the  exciting  coil  of 
the  single-phase  induction  motor.  In  the  direction  at  right 
angles  to  the  axis  of  the  exciting  coil,  however,  the  magnetic 
flux  is  due  to  the  M.M.F.  of  the  armature  currents  alone, 
no  primary  E.M.F.  acting  in  this  direction. 

Consequently,  in  the  polyphase  motor  running  light, 
that  is,  doing  no  work  whatever,  the  armature  becomes 
currentless,  and  the  primary  currents  are  the  exciting  cur- 
rent of  the  motor  only.  In  the  single-phase  induction  motor, 
even  when  running  light,  the  armature  still  carries  the  ex- 
citing current  of  the  magnetic  flux  in  quadrature  with  the 
axis  of  the  primary  exciting  coil.  Since  this  flux  has 
essentially  the  same  intensity  as  the  flux  in  the  direction  of 
the  axis  of  the  primary  exciting  coil,  the  current  in  the 
armature  of  the  single-phase  induction  motor  running  light, 
and  therefore  also  the  primary  current  corresponding  thereto> 


INDUCTION  MACHINES.  283 

has  the  same  M.M.F.,  that  is,  the  same  intensity  as  the 
primary  exciting  current,  and  the  total  primary  current  of 
the  single-phase  induction  motor  running  light  is  thus  twice 
the  exciting  current,  that  is,  it  is  the  exciting  current  of  the 
main  magnetic  flux  plus  the  current  inducing  in  the  armature 
the  exciting  current  of  the  cross  magnetic  flux.  In  the 
armature  or  secondary  at  synchronism  this  exciting  current 
is  a  current  of  twice  the  primary  frequency,  at  any  other 
speed  it  is  of  a  frequency  equal  to  speed  (in  cycles)  plus 
synchronism. 

Thus,  if  in  a  quarter-phase  motor  running  light  one  phase 
is  open  circuited,  the  current  in  the  other  phase  doubles.  If 
in  the  three-phase  motor  two  phases  are  open  circuited,  the 
current  in  the  third  phase  trebles,  since  the  resultant  M.M.F. 
of  a  three-phase  machine  is  1.5  times  that  of  one  phase. 
In  consequence  thereof,  the  total  volt-ampere  input  of  the 
motor  remains  the  same,  and  at  the  same  magnetic  density, 
or  the  same  impressed  E.M.F.,  all  induction  motors,  single- 
phase  as  well  as  polyphase,  consume  approximately  the  same 
volt-ampere  input,  and  the  same  power  input  for  excitation, 
and  give  the  same  distribution  of  magnetic  flux. 

Since  the  maximum  output  of  a  single-phase  motor  at 
the  same  impressed  E.M.F.  is  considerably  less  than  that  of 
a  polyphase  motor,  it  follows  therefrom  that  the  relative  ex- 
citing current  in  the  single-phase  motor  must  be  larger. 

The  cause  of  this  cross  magnetization  in  the  single-phase 
induction  motor  near  synchronism  is  that  the  induced  arma- 
ture currents  lag  90°  behind  the  inducing  magnetism,  and 
are  carried  by  the  synchronous  rotation  90°  in  space  before 
reaching  their  maximum,  thus  giving  the  same  magnetic 
effect  as  a  quarter-phase  E.M.F.  impressed  upon  the  primary 
system  in  quadrature  position  with  the  main  coil.  Hence 
they  can  be  eliminated  by  impressing  a  magnetizing  quadra- 
ture E.M.F.  upon  an  auxiliary  motor  circuit  as  is  done  in 
the  monocyclic  motor. 

Below  synchronism,  the  induced  armature  currents  are 


284  ELECTRICAL   ENGINEERING. 

carried  less  than  90°,  and  thus  the  cross  magnetization  due 
to  them  is  correspondingly  reduced,  and  becomes  zero  at 
standstill. 

The  torque  is  proportional  to  the  armature  energy  cur- 
rents times  the  intensity  of  magnetic  flux  in  quadrature 
position  thereto. 

In  the  single-phase  induction  motor,  the  armature  energy 
currents  //=  ea^  can  flow  only  coaxially  with  the  primary 
coil,  since  this  is  the  only  position  in  which  corresponding 
primary  currents  can  exist.  The  magnetic  flux  in  quadrature 
position  is  proportional  to  the  component  e  carried  in  quadra- 
ture, or  approximately  to  (1  —  s)  e,  and  the  torque  is  thus, 

T=  (1  -  j)  elr  =  (1  -  j)  ^ 

thus  decreases  much  faster  with  decreasing  speed,  and  be- 
comes zero  at  standstill.  The  power  is  then, 

P  =  (1  -  sfel'=  (1  -  j-)V^. 

Since  in  the  single-phase  motor  one  primary  circuit  only, 
but  a  multiplicity  of  secondary  circuits  exists,  all  secondary 
circuits  are  to  be  considered  as  corresponding  to  the  same 
.primary  circuit,  and  thus  the  joint  impedance  of  all  sec- 
ondary circuits  must  be  used,  as  the  secondary  impedance, 
at  least  at  or  near  synchronism.  Thus,  if  the  armature  has  a 
quarter-phase  winding  of  impedance  Zl  per  circuit,  the  resul- 
tant secondary  impedance  is  — L,  if  it  contains  a  three-phase 

2 

winding  of  impedance  Z1  per  circuit,  the  resultant  secondary 

.    Z. 

impedance  is  — -. 
o 

In  consequence  hereof  the  resultant  secondary  impe- 
dance of  a  single-phase  motor  is  less  in  comparison  with  the 
primary  impedance  than  in  the  polyphase  motor.  Since  the 
drop  of  speed  under  load  depends  upon  the  secondary  resist- 
ance, in  the  single-phase  induction  motor  the  drop  in  speed 
at  load  is  generally  less  than  in  the  polyphase  motor.  This 
greater  constancy  of  speed  of  the  single-phase  induction 


INDUCTION  MACHINES.  285 

motor  has  led  to  the  erroneous  opinion  that  such  a  motor 
operates  at  synchronism,  while  obviously,  just  as  the 
polyphase  induction  motor,  it  can  never  reach  complete 
synchronism. 

The  further  calculation  of  the  single-phase  induction 
motor  is  identical  with  that  of  the  polyphase  induction  motor, 
as  given  in  the  former  chapter. 

In  general,  no  special  motors  are  used  for  single-phase 
circuits,  but  polyphase  motors  adapted  thereto.  An  induc- 
tion motor  with  one  primary  winding  only  could  not  be 
started  by  a  phase-splitting  device,  and  would  necessarily 
be  started  by  external  means.  A  polyphase  motor,  as  for 
instance  a .  three-phase  motor  operating  single-phase,  by 
having  two  of  its  terminals  connected  to  the  single-phase 
mains,  is  just  as  satisfactory  a  single-phase  motor  as  one 
built  with  one  primary  winding  only.  The  only  difference 
is,  that  in  the  latter  case  a  part  of  the  circumference  of  the 
primary  structure  is  left  without  winding,  while  in  the  poly- 
phase motor  this  part  contains  windings  also,  which,  how- 
ever, are  not  used,  or  are  not  effective  when  running  as 
single-phase  motor,  but  are  necessary  when  starting  by 
means  of  displaced  E.M.F.'s.  Thus,  in  a  three-phase  motor 
operating  from  single-phase  mains,  in  starting,  the  third 
terminal  is  connected  to  a  phase-displacing  device,  giving 
to  the  motor  the  cross  magnetization  in  quadrature  to  the 
axis  of  the  primary  coil,  which  at  speed  is  produced  by  the 
rotation  of  the  induced  secondary  currents,  and  which  is 
necessary  for  producing  the  torque  by  its  action  upon  the 
induced  secondary  energy  currents. 

Thus  the  investigation  of  the  single-phase  induction 
motor  resolves  itself  into  the  investigation  of  the  polyphase 
motor  operating  on  single-phase  circuits. 


286  ELECTRICAL   ENGINEERING. 


2.    LOAD  AND  SPEED  CURVES. 

Comparing  thus  a  three-phase  motor  of  exciting  admit- 
tance per  circuit  F0  =  g  -rjb,  and  self-inductive  impedances 
Z^=  r0  —  _/r0,  and  Z±  =  rx  —  jx^  per  circuit,  with  the  same 
motor  operating  as  single-phase  motor  from  one  pdr  of 
terminals,  the  single-phase  exciting  admittance  is  F0'  =  3  Y 
(so  as  to  give  the  same  voltamperes  excitation  3^F0),  the 
primary  self -inductive  impedance  is  the  same,  Z0  =  r0  —  y!r0, 
the  secondary  self-inductive  impedance  single-phase,  how- 
ever, is  only  Z{  =  -1,  since  all  three  secondary  circuits  cor- 

o 

respond    to  the    same  primary   circuit,  and    thus  the  total 
impedance    single-phase  is  Z'—Z^-\-  — 1,  while  that  of  the 

o 

three-phase  motor  is  Z  =  Z0  +  Zr 

Assuming  approximately  Z0=  Zv  we  have, 


Thus,  absolute,  it  is, 


Z'  =  f  Z,  and 

ef  =  2  e. 


That  is,  the  characteristic  constant  of  a  motor  running 
single-phase  is  twice  what  it  is  running  three-phase,  or  poly- 
phase in  general. 

Hence,  the  ratio  of  exciting  current  to  current  at  stand- 
still, or  of  waste  flux  to  useful  flux,  is  doubled  by  changing 
from  polyphase  to  single-phase. 

This  explains  the  inferiority  of  the  single-phase  motor 
compared  with  the  polyphase  motor. 

As  a  rule,  an  average  polyphase  motor  makes  a  poor 
single-phase  motor,  and  a  good  single-phase  motor  must  be 
an  excellent  polyphase  motor. 

As  instances  are  shown  in  Figs.  130  and  131  the  load 
curves  and  speed  curves  of  the  three-phase  motor  of  which 


IND  UC  TION  MA  CHINES. 


287 


the  curves  of  one  circuit  are  given  in  Figs.  125  and  126,  of 
the  constants  : 

-THREE-PHASE,  ea  •=.  110.  SINGLE-PHASE. 

yo  =  .01  +  .!/,  Y0  =  .03  +  .3/, 


Thus,    0  =  6.36. 


Z,  =  .033  -  .!/, 
Thus,  0  =  12.72. 


737. 


288  .  ELECTRICAL   ENGINEERING. 

It  is  of  interest  to  compare  Fig.  130  with  Fig-.  125  and 
to  note  the  lesser  drop  of  speed  (due  to  the  relatively  lower 
secondary  resistance)  and  lower  power  factor  and  efficiencies, 
especially  at  light  load.  The  maximum  output  is  reduced 
from  3  X  7000  =  21,000  in  the  three-phase  motor  to  9100 
watts  in  the  single-phase  motor. 

Since,  however,  the  internal  losses  are  less  in  the  single- 
phase  motor,  it  can  be  operated  at  from  25  to  30  $  higher 
magnetic  density  than  the  same  motor  polyphase,  and  in 
this  case  its  output  is  from  f  to  f  that  of  the  polyphase 
motor. 

The  preceding  discussion  of  the  single-phase  induction 
motor  is  approximate,  and  correct  only  at  or  near  synchro- 
nism, where  the  magnetic  field  is  practically  a  uniformly 
rotating  field  of  constant  intensity,  that  is,  the  quadrature 
flux  produced  by  the  armature  magnetization  equal  to  the 
main  magnetic  flux  produced  by  the  impressed  E.M.F. 

If  an  accurate  calculation  of  the  motor  at  intermediate 
speed  and  at  standstill  is  required,  the  change  of  effective 
exciting  admittance  and  of  secondary  impedance,  due  to  the 
decrease  of  the  quadrature  flux,  have  to  be  considered. 

At  synchronism  the  total  exciting  admittance  gives  the 
M.M.F.  of  main  flux  and  auxiliary  flux,  while  at  standstill 
the  quadrature  flux  has  disappeared  or  decreased  to  that 
given  by  the  starting  device,  and  thus  the  total  exciting  ad- 
mittance has  decreased  to  one-half  of  its  synchronous  value, 
or  one-half  plus  the  exciting  admittance  of  the  starting  flux. 

The  effective  secondary  impedance  at  synchronism  is  the 
joint  impedance  of  all  secondary  circuits,  at  standstill,  how- 
ever, only  the  joint  impedance  of  the  projection  of  the 
secondary  coils  on  the  direction  of  the  main  flux,  that  is, 
twice  as  large  as  at  synchronism.  In  other  words,  from 
standstill  to  synchronism  the  effective  secondary  impe- 
dance gradually  decreases  to  one-half  its  standstill  value  at 
synchronism. 

For  fuller  discussion  hereof  the  reader  must  be  referred 


2ND  UC  TION  MA  CHINES. 


289 


to  my  second  paper  on  the  Single-phase  Induction  Motor, 
Transactions  A.  I.  E.  E.,  1900,  page  37. 

The  torque  in  Eig.  131  obviously  slopes  towards  zero  at 
standstill.  The  effect  of  resistance  inserted  in  the  secondary 
of  the  single-phase  motor  is  similar  as  in  the  polyphase  motor 
in  so  far  as  an  increase  of  resistance  lowers  the  speed  at 
which  the  maximum  torque  takes  place.  While,  however, 
in  the  polyphase  motor  the  maximum  torque  remains  the 
same,  and  merely  shifts  towards  lower  speed  with  the  increase 
of  resistance,  in  the  single-phase  motor  the  maximum  torque 
decreases  proportionally  to  the  speed  at  which  the  maximum 
torque  point  occurs,  due  to  the  factor  (1  -  s)  entering  the 
equation  of  the  torque, 

T=  ^(1  —  s). 

Thus,  in  Fig.  132  are  given  the  values  of  torque  of  the 
single-phase  motor  for  the  same  conditions  and  the  same 
motor  of  which  the  speed  curves  polyphase  are  given  in 
Fig.  128. 


THREE-PHASE  MOTOR 


Z0=Z,T.1-.3JT 
_ON.S  N.GLE^EHAS.E_ClR.C.UJiIL 


oco 

000 


Fig.  132. 


The  maximum  value  of  torque  which  can  be  reached  at 
any  speed  lies  on  the  tangent  drawn  from  the  origin  on  to 
the  torque  curve  for  r^—  .1  or  short  circuited  secondary.  At 
low  speeds  the  torque  of  the  single-phase  motor  is  greatly 


290  ELECTRICAL   ENGINEERING 

increased  by  the  insertion  of  secondary  resistance,  just  as  in 
the  polyphase  motor. 

3.   STARTING  DEVICES  OF  SINGLE-PHASE  MOTORS. 

At  standstill,  the  single-phase  induction  motor  has  no 
starting  torque,  since  the  line  of  polarization  due  to  the 
armature  currents  coincides  with  the  axis  of  magnetic  flux 
impressed  by  the  primary  circuit.  Only  when  revolving, 
torque  is  produced,  due  to  the  axis  of  armature  polarization 
being  shifted  by  the  rotation,  against  the  axis  of  magnetism, 
until  at  or  near  synchronism  it  is  in  quadrature  therewith, 
and  the  magnetic  disposition  thus  identical  with  that  of  the 
polyphase  induction  motor. 

Leaving  out  of  consideration  starting  by  mechanical 
means,  and  starting  by  converting  the  motor  into  a  series  or 
shunt  motor,  that  is,  by  passing  the  alternating  current  by 
means  of  commutator  and  brushes  through  both  elements  of 
the  motor,  the  following  methods  of  starting  single-phase 
motors  are  left  : 

1st.  Shifting  of  the  axis  of  armature  or  secondary  polari- 
zation against  the  axis  of  inducing  magnetism. 

2d.  Shifting  the  axis  of  magnetism,  that  is,  producing  a 
magnetic  flux  displaced  in  position  from  the  flux  inducing 
the  armature  currents. 

The  first  method  requires  a  secondary  system  which  is 
unsymmetrical  in  regard  to  the  primary,  and  thus,  since  the 
secondary  is  movable,  requires  means  of  changing  the 
secondary  circuit,  such  as  commutator  brushes  short  circuit- 
ing secondary  coils  in  the  position  of  effective  torque,  and 
open  circuiting  them  in  the  position  of  opposing  torque. 

Thus  this  method  leads  to  the  repulsion  motor,  which  is 
a  commutator  motor  also. 

With  the  commutatorless  induction  motor,  or  motor  with 
permanently  closed  armature  circuits,  all  starting  devices 
consist  in  establishing  an  auxiliary  magnetic  flux  in  phase 


INDUCTION  MACHINES.  291 

with  the  induced  secondary  currents  in  time,  and  in  quadra- 
ture with  the  line  of  armature  polarization  in  space.  They 
consist  in  producing  a  component  of  magnetic  flux  in  quad- 
rature in  space  with  the  primary  magnetic  flux  inducing  the 
armature  currents,  and  in  phase  with  the  latter ;  that  is,  in 
quadrature  with  the  primary  magnetic  flux. 
Thus  if, 

P  =  polarization  due  to  the  induced  or  armature  currents, 
M  =  auxiliary  magnetic  flux, 

<£  =  phase  displacement  in  time  between  M  and  P, 
and, 

w  =  phase  displacement  in  space  between  M  and  Pt 

the  torque  is, 

T  =  PM  sin  <o  cos  <j>. 

In  general  the  starting  torque,  apparent  torque  efficiency, 
etc.,  of  the  single-phase  induction  motor  with  any  of  these 
devices  are  given  in  percent  of  the  corresponding  values  of 
the  same  motor  with  polyphase  magnetic  flux,  that  is,  with  a 
magnetic  system  consisting  of  two  equal  magnetic  fluxes  in 
quadrature  in  time  and  space. 

The  infinite  variety  of  arrangements  proposed  for  start- 
ing single-phase  induction  motors  can  be  grouped  into  three 
classes. 

1.  Phase- Splitting  Devices.  The  primary  system  is 
composed  of  two  or  more  circuits  displaced  from  each  other 
in  position,  and  combined  with  impedances  of  different  in- 
ductance factors  so  as  to  produce  a  phase  displacement 
between  them. 

When  using  two  motor  circuits,  they  can  either  be  con- 
nected in  series  between  the  single-phase  mains,  and  shunted 
with  impedances  of  different  inductance  factors,  as,  for  in- 
stance, a  condensance  and  an  inductance,  or  they  can  be 
connected  in  shunt  between  the  single-phase  mains  but  in 
series  with  impedances  of  different  inductance  factors.  Ob- 
viously the  impedances  used  for  displacing  the  phase  of  the 


292  ELECTRICAL   ENGINEERING. 

exciting  coils  can  either  be  external  or  internal,  as  represented 
by  high-resistance  winding  in  one  coil  of  the  motor,  etc. 

In  this  class  belongs  the  use  of  the  transformer  as  a 
phase- splitting  device,  by  inserting  a  transformer  primary  in 
series  to  one  motor  circuit  in  the  main  line,  and  connecting 
the  other  motor  circuit  to  the  secondary  of  the  transformer, 
or  by  feeding  one  of  the  motor  circuits  directly  from  the 
mains,  and  the  other  from  the  secondary  of  a  transformer 
connected  across  the  mains  with  its  primary.  In  either 
case,  it  is  the  internal  impedance,  respectively  internal  admit- 
tance, of  the  transformer  which  is  combined  with  one  of  the 
motor  circuits  for  displacing  its  phase,  and  thus  this  arrange- 
ment becomes  most  effective  by  using  transformers  of  high 
internal  impedance  or  admittance,  as  constant  power  trans- 
formers or  open  magnetic  circuit  transformers. 

2.  Inductive    Devices.     The  motor   is    excited  by  the 
combination  of  two  or  more  circuits  which  are  in  inductive 
relation  to  each  other.     This  mutual  induction  between  the 
motor  circuits  can  either  take  place  outside  of  the.  motor  in 
a  separate  phase-splitting  device,  or  in  the  motor  proper. 

In  the  first  case  the  simplest  form  is  the  divided  circuit 
whose  branches  are  inductively  related  to  each  other  by 
passing  around  the  same  magnetic  circuit  external  to  the 
motor. 

In  the  second  case  the  simplest  form  is  the  combination 
of  a  primary  exciting  coil  and  a  short  circuited  secondary  coil 
induced  thereby  on  the  primary  member  of  the  motor,  or  a 
secondary  coil  closed  by  an  impedance. 

In  this  class  belong  the  use  of  the  shading  coil  and  the 
accelerating  coil. 

3.  Monocyclic  Starting  Devices.     An    essentially  watt- 
less E.M.F.  of  displaced  phase  is  produced  outside  of  the 
motor,  and  used  to  energize  a  cross  magnetic  circuit  of  the 
motor,  either  directly  by  a  special  teaser  coil  on  the  motor, 
or  indirectly  by  combining  this  wattless    E.M.F.   with  the 
main   E.M.F. and    thereby  deriving   a  system  of    E.M.F.'s 


INDUCTION  MACHINES.  293 

of  approximately  three-phase  or  any  other  relation.  In  this 
case  the  primary  system  of  the  motor  is  supplied  essentially 
by  a  polyphase  system  of  E.M.F.  with  a  single-phase  flow 
of  energy,  a  system,  which  I  have  called  "monocyclic." 

For  a  complete  discussion  and  theoretical  investigation 
of  the  different  starting  devices,  the  reader  must  be  referred 
to  the  paper  on  the  single-phase  induction  motor,  "  Ameri- 
can Institute  of  Electrical  Engineers'  Transactions,  1898," 
February. 

The  use  of  the  resistance-inductance,  or  monocyclic 
starting  device  with  three-phase  wound  induction  motor, 
will  be  discussed  somewhat  more  explicitly  as  the  only 
method  not  using  condensers,  which  has  found  extensive 
commercial  application.  It  gives  relatively  the  best  starting 
torque  and  torque  efficiencies. 

In  Fig.  133,  M  represents  a  three-phase  induction  motor 
of  which  two  terminals,  1  and  2,  are  connected  to  single- 


Fig.  133. 

phase  mains,  and  the  terminal  3  to  the  common  connection 
of  a  conductance  a  (that  is,  a  resistance  -I  and  an  equal 

susceptance  ja  (thus  a  reactance  —  -J  connected  in  series 

across  the  mains. 

Let    Y  =  g  +  jb  =  total  admittance    of    motor  between 
terminals  1  and  2  while  at  rest.     We  then  have, 

4  Y  =  total  admittance  from  terminal  3  to  terminals  1 
and  2,  regardless  whether  the  motor  is  delta  or  Y  wound. 
Let  e  =  E.M.F.  in  single-phase  mains. 

E  =  difference  of  potential  across  conductance  a  of  the 
starting  device. 


294  ELECTRICAL   ENGINEERING. 

We  then  have, 

Current  in  a,  7X  =  £a, 

E.M.F.  across  /#,         e  —  E. 

Thus,  current  in  ja, 

&=ja(e  -  E\ 

Thus  current  in  cross  magnetizing  motor  circuit  from  3  to 
1,2, 

f,  =  L-L  =  Ea-ja(e-E}. 

The  E.M.F.  of  the  cross  magnetizing  circuit  is,  as  may 
be  seen  from  the  diagram  of  E.M.F.'s,  which  form  a  tri- 
angle with  EQ  E  and  e  —  E  as  sides, 

£Q  =  £-(t-£)  =  2£-t, 
and  since, 

&^i*£o, 

we  have, 


This  expression  solved  for  E  becomes, 

3>-4F 


which  from  the  foregoing  value  of  EQ  gives, 

3^(y-l) 
~  3  a  +  3ja  -  8  Y  ' 
or,  substituting, 

?.•=•**.  A 

expanding,   and  multiplying  both    numerator   and    denomi 
nator  by 

(3a-  Sg)  -  j(Za  -  8^), 


and  the  imaginary  component  thereof,  or  E.M.F.  in  quadra- 
ture to  e  in  time  and  in  space,  is 

£J =-Je<*7 — 


INDUCTION  MACHINES.  295 

In  the  same  motor  on  three-phase  circuit  this  quadrature 
E.M.F.  is  the  altitude  of  the  equilateral  triangle  with  e  as 

V3 
sides,  thus  =^>-r- ,  and  since  the   starting  torque  of   the 

motor  is  proportional  to  this  quadrature  E.M.F.,  the  relative 
starting  torque  of  the  monocyclic  starting  device,  or  the 
ratio  of  starting  torque  of  the  motor  with  monocyclic  start- 
ing device  that  of  the  same  motor  on  three-phase  circuit,  is 


/  = 


2 

For  further  discussion  of  this  subject  the  reader  is 
referred  to  the  paper  on  "  Single-phase  Induction  Motors  " 
mentioned  above. 

4.    ACCELERATION  WITH  STARTING  DEVICE. 

The  torque  of  the  single-phase  induction  motor  (without 
starting  device)  is  proportional  to  the  product  of  main  flux 
or  magnetic  flux  produced  by  the  primary  impressed  E.M.F., 
and  the  speed.  Thus  it  is  the  same  as  in  the  polyphase 
motor  at  or  very  near  synchronism,  but  falls  off  with  decreas- 
ing speed  and  becomes  zero  at  standstill. 

To  produce  a  starting  torque,  a  device  has  to  be  used  to 
impress  an  auxiliary  magnetic  flux  upon  the  motor,  in  quad- 
rature with  the  main  flux  in  time  and  in  space,  and  the  start- 
ing torque  is  proportional  to  this  auxiliary  or  quadrature 
flux.  During  acceleration  or  at  intermediate  speed,  the  torque 
of  the  motor  is  the  resultant  of  the  main  torque,  or  torque 
produced  by  the  primary  main  flux,  and  the  auxiliary 
torque  produced  by  the  auxiliary  quadrature  or  starting  flux. 
In  general,  this  resultant  torque  is  not  the  sum  of  main  and 
auxiliary  torque,  but  less,  due  to  the  interaction  between 
the  motor  and  the  starting  device. 

All  the  starting  devices  depend  more  or  less  upon  the 
total  admittance  of  the  motor  and  its  power  factor.  With 


296  ELECTRICAL   ENGINEERING. 

increasing  speed,  however,  the  total  admittance  of  the  motor 
decreases,  and  its  power  factor  increases,  and  an  auxiliary 
torque  device  suited  for  the  admittance  of  the  motor  at 
standstill  will  not  be  suited  for  the  changed  admittance  at 
speed. 

The  currents  induced  in  the  secondary  by  the  main  or 
primary  magnetic  flux  are  carried  by  the  rotation  of  the 
motor  more  or  less  in  quadrature  position,  and  thus  produce 
the  quadrature  flux  giving  the  main  torque  as  discussed 
before. 

This  quadrature  component  of  the  main  flux  induces  an 
E.M.F.  in  the  auxiliary  circuit  of  the  starting  device,  and 
thus  changes  the  distribution  of  currents  and  E.M.F.'s  in 
the  starting  device.  The  circuits  of  the  starting  device 
then  contain,  besides  the  motor  admittance  and  external 
admittance,  an  active  counter  E.M.F. ,  changing  with  the 
speed.  Inversely,  the  currents  produced  by  the  counter 
E.M.F.  of  the  motor  in  the  auxiliary  circuit  react  upon  the 
counter  E.M.F.,  that  is,  upon  the  quadrature  component  of 
main  flux,  and  change  it. 

Thus  during  acceleration  we  have  to  consider  : 

1.  The   effect  of  the  change  of  total  motor  admittance, 
and  its  power  factor,  upon  the  starting  device. 

2.  The  effect  of  the  counter  E.M.F.  of  the  motor  upon 
the  starting  device,  and  the   effect   of  the  starting  device 
upon  the  counter  E.M.F.  of  the  motor. 

1.  The  total  motor  admittance  and  its  power  factor 
change  very  much  during  acceleration  in  motors  with  short 
circuited  low  resistance  secondary.  In  such  motors  the 
admittance  at  rest  is  very  large  and  its  power  factor  low, 
and  with  increasing  speed  the  admittance  decreases  and  its 
power  factor  increases  greatly.  In  motors  with  short  cir- 
cuited high  resistance  secondary,  the  admittance  also  de- 
creases greatly  during  acceleration,  but  its  power  factor 
changes  less,  being  already  high  at  standstill.  Thus  the 
starting  device  will  be  affected  less.  Such  motors,  however, 


INDUCTION  MACHINES.  297 

are  inefficient  at  speed.  In  motors  with  variable  secondary 
resistance  the  admittance  and  its  power  factor  can  be  main- 
tained constant  during  acceleration,  by  decreasing  the  resist- 
ance of  the  secondary  circuit  in  correspondence  with  the 
increasing  counter  E.M.F.  Hence,  in  such  motors  the  start- 
ing device  is  not  thrown  out  of  adjustment  by  the  changing 
admittance  during  acceleration. 

In  the  phase-splitting  devices,  and  still  more  in  the 
inductive  devices,  the  starting  torque  depends  upon  the 
internal  or  motor  admittance,  and  is  thus  essentially  affected 
by  the  change  of  admittance  during  acceleration,  and  by  the 
appearance  of  a  counter  E.M.F.  during  acceleration,  which 
throws  the  starting  device  out  of  its  proper  adjustment,  so 
that  frequently  while  a  considerable  torque  exists  at  stand- 
still, this  torque  becomes  zero  and  then  reverses  at  some 
intermediate  speed,  and  the  motor,  while  starting  with  fair 
torque,  is  not  able  to  run  up  to  speed  with  the  starting 
device  in  circuit.  Especially  is  this  the  case  where  capacity 
is  used  in  the  starting  device. 

IV.   Induction  Generator. 
1.    INTRODUCTION. 

In  the  range  of  slip  from  s  =  0  to  s  =  1,  that  is,  from 
synchronism  to  standstill,  torque,  power  output,  and  power 
input  of  the  induction  machine  are  positive,  and  the  machine 
thus  acts  as  a  motor,  as  discussed  before. 

Substituting,  however,  in  the  equations  in  paragraph  1 
for  s  values  >  1,  corresponding  to  backward  rotation  of  the 
machine,  the  power  input  remains  positive,  the  torque  also 
remains  positive,  that  is,  in  the  same  direction  as  for  s  <  1, 
but  since  the  speed  (1  —  s)  becomes  negative  or  in  opposite 
direction,  the  power  output  is  negative,  that  is,  the  torque 
in  opposite  direction  to  the  speed.  In  this  case,  the  machine 
consumes  electrical  energy  in  its  primary  and  mechanical 


298 


ELECTRICAL   ENGINEERING. 


energy  by  a  torque  opposing  the  rotation,  thus  acting  as 
brake. 

The  total  power,  electrical  as  well  as  mechanical,  is  con- 
sumed by  internal  losses  of  the  motor.  Since,  however, 
with  large  slip  in  a  low  resistance  motor,  the  torque  and 
power  are  small,  the  braking  power  of  the  induction  machine 
at  backward  rotation  is,  as  a  rule,  not  considerable. 

Substituting  for  s  negative  values,  corresponding  to  a 
speed  above  synchronism,  torque  and  power  output  and 
power  input  become  negative,  and  a  load  curve  can  be 
plotted  for  the  induction  generator  which  is  very  similar, 
but  the  negative  counterpart  of  the  induction  motor  lead 
curve.  It  is  for  the  machine  shown  as  motor  in  Fig.  125, 
given  as  Fig.  134,  while  Fig.  135  gives  the  complete  speed 
curve  of  this  machine  from  s  =  1.5  to  s  =  —  1. 


Fig.  134. 


The  generator  part  of  the  curve  f or  s  <  0  is  of  the  same 
character  as  the  motor  part,  s  >  0,  but  the  maximum  torque 
and  maximum  output  of  the  machine  as  generator  are  greater 
than  as  motor. 


IATD  UC  TION  MA  CHINES. 


299 


Thus  an  induction  motor  when  speeded  up  above  syn- 
chronism acts  as  powerful  brake  by  returning  energy  into 
the  lines,  and  the  maximum  braking  effect  or  the  maximum 
energy  returned  by  the  machine  will  be  greater  than  the 
maximum  motor  torque  or  output. 


TORQUE 

3000 


6000 
4000 


I  1 

bUCmON    MACHINE 


ggss-- 


CONSTANT  TERMINAL 
VOLTAGE  OF  110- 


-go™ 


CRATOR 


SYNCHRONISM 


Je     Ja     .|7    ..      ..      .|»   . 


Fig.  135. 


2.    CONSTANT  SPEED    INDUCTION    OR   ASYNCHRONOUS 
GENERATOR. 

The  curves  in  Fig.  10  are  calculated  at  constant  fre- 
quency TV7",  and  thus  to  vary  the  output  of  the  machine  as 
generator  the  speed  has  to  be  increased.  This  condition 
may  be  realized  in  case  of  induction  generators  running  in 
parallel  with  synchronous  generators  under  conditions  where 
it  is  desirable  that  the  former  should  take  as  much  load  as 
its  driving  power  permits  ;  as,  for  instance,  if  the  induction 
generator  is  driven  by  a  water  power  while  the  synchronous 
generator  is  driven  by  steam  engine.  In  this  case,  the  control 
of  speed  would  be  affected  on  the  synchronous  generator, 
and  the  asynchronous  generator  be  without  speed-controlling 


300 


ELECTRICAL   ENGINEERING. 


devices,  running  up  beyond  synchronous  speed  as  much  as 
required  to  consume  the  power  supplied  to  it. 

Conversely,  however,  if  an  induction  machine  is  driven  at 
constant  speed  and  connected  to  a  suitable  circuit  as  load, 
the  frequency  given  by  the  machine  will  not  be  synchronous 
with  the  speed  or  constant  at  all  loads,  but  decreases  with 
increasing  load  from  practical  synchronism  at  no  load,  and 
thus  for  the  induction  generator  at  constant  speed  a  load 
curve  can  be  constructed  as  shown  in  Fig.  136,  giving  the 


Fig.  136. 


decrease  of  frequency  with  increasing  load  in  the  same  man- 
ner as  the  speed  of  the  induction  motor  at  constant  fre- 
quency decreases  with  the  load.  In  the  calculation  of  these 
induction  generator  curves  for  constant  speed,  the  change  of 
frequency  with  the  load  has  obviously  to  be  considered,  that 
is,  in  the  equations  the  reactance  ;trn  has  to  be  replaced  by  the 
reactance  ;r0  (1  —  s),  otherwise  the  equations  remain  the 
same. 


IND  UC  TION  MA  CHINES. 


301 


3.  POWER  FACTOR  OF  INDUCTION  GENERATOR. 

The  induction  generator  differs  essentially  from  a  syn- 
chronous alternator  (that  is,  a  machine  in  which  an  armature 
revolves  relatively  through  a  constant  or  continuous  mag- 
netic field)  by  having  a  power  factor  requiring  leading  cur- 
rent. That  is,  in  the  synchronous  alternator  the  phase 
relation  between  current  and  terminal  voltage  depends  en- 
tirely upon  the  external  circuit,  and  according  to  the  nature 
of  the  circuit  connected  to  the  synchronous  alternator  the 
current  can  lag  or  lead  the  terminal  voltage  or  be  in  phase 
therewith.  In  the  induction  or  asynchronous  generator, 
however,  the  current  must  lead  the  terminal  voltage  by  the 
angle  corresponding  to  the  load  and  voltage  of  the  machine, 
or  in  other  words  the  phase  relation  between  current  and 
voltage  in  the  external  circuit  must  be.  such  as  required  by 
the  induction  generator  at  that  particular  load. 

Induction  generators  can  operate  only  on  circuits  with 
leading  current  or  circuits  of  negative  effective  reactance. 

In  Fig.  137  are  given  for  the  constant  speed  induction 
generator  in  Fig.  136  as  function  of  the  impedance  of  the 


Toy^ 


E  I^DU<jrriCjN   GENEJRAT|OR 
-Y0=.01   +.1J 

^^ 


R   FACTOR 


E  FACT 


EXTERNAL 


CIRCUIT 


£ 


MPEDANCE,  Z 

"OK  MS" 
F/g.  737. 


302  ELECTRICAL   ENGINEERING. 

external  circuit  z  =  -2  as  abscissae  (where  e0  =  terminal  volt- 

zo 

age,  z'0  =  current  in  external  circuit),  the  leading  power  factor 
p  =  cos  w  required  in  the  load,  the  inductance  factor  q  = 
sin  <o,  and  the  frequency. 

Hence,  when  connected  to  a  circuit  of  impedance  2  this 
induction  generator  can  operate  only  if  the  power  factor  of 
its  circuit  is/  ;  and  if  this  is  the  case  the  voltage  is  indefinite, 
that  is,  the  circuit  unstable,  even  neglecting  the  impossibility 
of  securing  exact  equality  of  the  power  factor  of  the  external 
circuit  with  that  of  the  induction  generator. 

Two  possibilities  thus  exist  with  such  an  induction 
generator  circuit. 

1st.  The  power  factor  of  the  external  circuit  is  constant 
and  independent  of  the  voltage,  as  when  the  external  circuit 
consists  of  resistances,  inductances,  and  capacities. 

In  this  case  if  the  power  factor  of  the  external  circuit  is 
higher  than  that  of  the  induction  generator,  that  is,  the  lead- 
ing current  less,  the  induction  generator  fails  to  excite  and 
generate.  If  the  power  factor  of  the  external  circuit  is 
lower  than  that  of  the  induction  generator,  the  latter  excites 
and  its  voltage  rises  until  by  saturation  of  its  magnetic  cir- 
cuit and  the  consequent  increase  of  exciting  admittance, 
that  is,  decrease  of  internal  power  factor,  its  power  factor  has 
fallen  to  equality  with  that  of  the  external  circuit. 

In  this  respect  the  induction  generator  acts  like  the 
direct  current  shunt  generator,  that  is,  it  becomes  stable 
only  at  saturation,  but  loses  its  excitation  and  thus  drops 
its  load  as  soon  as  the  voltage  falls  below  saturation. 

Since,  however,  the  field  of  the  induction  generator  is 
alternating,  it  is  usually  not  feasible  to  run  at  saturation, 
due  to  excessive  hysteresis  losses,  except  for  very  low  fre- 
quencies. 

2d.  The  power  factor  of  the  external  circuit  depends 
upon  the  voltage  impressed  upon  it. 

This,  for  instance,  is  the  case  if  the  circuit  consists  of  a 


INDUCTION  MACHINES.  303 

synchronous  motor  or  contains  synchronous  motors  or    syn- 
chronous converters. 

In  the  synchronous  motor  the  current  is  in  phase  with 
the  impressed  E.M.F.  if  the  impressed  E.M.F.  equals  the 
counter  E.M.F.  of  the  motor,  plus  the  internal  loss  of  volt- 
age. It  is  leading  if  the  impressed  E.M.F.  is  less,  and 
lagging  if  the  impressed  E.M.F.  is  more.  Thus  when  con- 
necting an  induction  generator  with  a  synchronous  motor,  at 
constant  field  excitation  of  the  latter  the  voltage  of  the  in- 
duction generator  rises  until  it  is  as  much  below  the  counter 
E.M.F.  of  the  synchronous  motor  as  required  to  give  the 
leading  current  corresponding  to  the  power  factor  of  the 
generator.  Thus  a  system  consisting  of  a  constant  speed 
induction  generator  and  a  synchronous  motor  at  constant 
field  excitation  is  absolutely  stable.  At  constant  field  exci- 
tation of  the  synchronous  motor,  at  no-load  the  synchronous 
motor  runs  practically  at  synchronism  with  the  induction 
generator,  with  a  terminal  voltage  slightly  below  the  coun- 
ter E.M.F.  of  the  synchronous  motor.  With  increase  of 
load  the  frequency  and  thus  the  speed  of  the  synchronous 
motor  drops,  due  to  the  slip  of  frequency  in  the  induction 
generator,  and  the  voltage  drops,  due  to  the  increase  of  lead- 
ing current  required,  and  the  decrease  of  counter  E.M.F. 
caused  by  the  decrease  of  frequency. 

By  increasing  the  field  excitation  of  the  synchronous 
motor  with  increase  of  load,  obviously  the  voltage  can  be 
maintained  constant,  or  even  increased  with  the  load. 

When  running  from  an  induction  generator  a  synchro- 
nous motor  gives  a  load  curve  very  similar  to  the  load  curve 
of  an  induction  motor  running  from  a  synchronous  generator  ; 
that  is,  a  magnetizing  current  at  no-load  and  a  speed  gradu- 
ally decreasing  with  the  increase  of  load  up  to  a  maximum 
output  point,  at  which  the  speed  curve  bends  sharply  down, 
the  current  curve  upward,  and  the  motor  drops  out  of  step. 

The  current,  however,  in  the  case  of  the  synchronous 
motor  operated  from  an  induction  generator  is  leading,  while 


304 


ELECTRICAL   ENGINEERING. 


it  is  lagging  in  an  induction  motor  operated  from  a  synchro- 
nous generator. 


Fig.  138. 

In  Fig.  138  is  shown  the  load  curve  of  a  synchronous 
motor  operated  from  the  induction  generator  in  Fig.  136. 

In  Fig.  139  is  shown  the  load  curve  of  an  over-com- 
pounded synchronous  converter  operated  from  an  induction 


INDUCTION  MACHINES. 


305 


generator,   the    over-compounding    being   such  as    to  give 
approximately  constant  terminal  voltage,  e. 

Obviously  when  operating  a  self-exciting  synchronous 
converter  from  an  induction  generator,  the  system  is  unstable 
also,  if  both  machines  are  below  magnetic  saturation,  since 
in  this  case  in  both  machines  the  induced  E.M.F.  is  propor- 


INDUCTION   GENERATOR 
\ND 


JUCTION 

_J 


ROTARY  CONVERTER 


PHASE   CONTROL. 
NO  LINE  IMPEDANCE. 


7 


.9        1  O      1 


Fig.  139. 

tional  to  the  field  excitation  and  the  field  excitation  pro- 
portional to  the  voltage.  That  is,  with  an  unsaturated 
induction  generator  the  synchronous  converter  operated 
therefrom  must  have  its  magnetic  field  excited  to  a  density 
above  the  bend  of  the  saturation  curve. 

Since  the  induction  generator  requires  for  its  Operation 
leading  current  varying  with  the  load  in  the  manner  deter- 
mined by  the  internal  constants  of  the  motor,  to  make  an 
induction  or  synchronous  generator  suitable  for  operation  on 
a  general  alternating  current  circuit,  it  is  necessary  to  have 


306  ELECTRICAL   ENGINEERING. 

a  synchronous  machine  as  exciter  in  the  circuit  supplying  the 
required  leading  current  to  the  induction  generator  ;  and  in 
this  case  the  voltage  of  the  system  is  controlled  by  the  field 
excitation  of  the  synchronous  machine,  that  is,  its  counter 
E.M.F.  Either  a  synchronous  motor  of  suitable  size  running 
light  can  be  used  herefor  as  exciter  of  the  induction  gen- 
erator, or  the  leading  exciting  current  of  the  induction 
generator  may  be  derived  from  synchronous  motors  or  con- 
verters in  the  same  system,  or  from  synchronous  alternating 
current  generators  operated  in  parallel  with  the  induction 
generator,  in  which  latter  case,  however,  these  currents  come 
from  the  synchronous  alternator  as  lagging  currents.  Elec- 
trostatic condensers  may  also  be  used  for  excitation,  but  in 
this  case  besides  the  condensers  a  synchronous  machine  is 
required  to  secure  stability. 

Therefore,  induction  generators  are  more  suited  for  cir- 
cuits which  normally  carry  leading  currents,  as  synchronous 
motor  and  synchronous  converter  circuits,  but  less  suitable 
for  circuits  with  lagging  currents,  since  in  the  latter  case 
an  additional  synchronous  machine  is  required,  giving  all  the 
lagging  currents  of  the  system  plus  the  induction  generator 
exciting  current. 

Obviously,  when  running  induction  generators  in  parallel 
with  a  synchronous  alternator,  no  synchronizing  is  required, 
but  the  induction  generator  takes  a  load  corresponding  to 
the  excess  of  its  speed  over  synchronism,  or  conversely  if  the 
driving  power  behind  the  induction  generator  is  limited,  no 
speed  regulation  is  required,  but  the  induction  generator  runs 
at  a  speed  exceeding  synchronism  by  the  amount  required 
to  consume  the  driving  power. 

The  foregoing  consideration  t  obviously  applies  to  the 
polyphase  induction  generator  as  well  as  the  single-phase 
induction  generator,  the  latter,  however,  requiring  a  larger 
exciter  in  consequence  of  its  lower  power  factor.  The 
curves  shown  in  the  preceding  apply  to  the  machine  as 
polyphase  generator. 


INDUCTION  MACHINES.  307 

The  effect  of  resistance  in  the  secondary  is  essentially 
the  same  in  the  induction  generator  as  in  the  induction  motor. 
An  increase  of  resistance  increases  the  slip,  that  is,  requires 
an  increase  of  speed  at  the  same  torque,  current  and  output, 
and  thus  correspondingly  lowers  the  efficiency. 

Regarding  the  synchronous  induction  machine,  that  is, 
a  machine  having  a  single-phase  or  polyphase  primary  and  a 
single-phase  secondary,  I  must  refer  to  my  book,  "  Theory 
and  Calculation  of  Alternating  Current  Phenomena,"  where 
it  is  shown  that  such  a  machine  not  only  operates  like  an 
ordinary  induction  machine  as  motor  below  and  as  generator 
above  synchronism,  but  can  operate  also  at  synchronism 
either  as  generator  or  as  motor,  according  to  the  phase 
relation  between  the  impressed  E.M.F.  and  the  position  of 
the  secondary  circuit,  the  work  being  done  in  a  similar 
manner  as  in  a  reaction  machine  by  a  distortion  of  the  wave 
shape,  or  what  may  be  called  an  energy  component  of  self- 
induction,  due  to  a  periodic  variation  of  inductance. 

V.  Induction  Booster. 

In  the  induction  machine,  at  a  given  slip  s,  current  and 
terminal  voltage  are  proportional  to  each  other  and  of  con- 
stant phase  relation,  and  their  ratio  is  a  constant.  Thus 
when  connected  in  an  alternating  circuit,  whether  in  shunt 
or  in  series,  and  held  at  a  speed  giving  a  constant  and  definite 
slip  s,  either  positive  or  negative,  the  induction  machine  acts 
like  a  constant  impedance. 

The  apparent  impedance  and  its  components,  the  apparent 
tresistance  and  apparent  reactance  represented  by  the  induc- 
tion machine,  vary  with  the  slip.  At  synchronism  apparent 
impedance,  resistance,  and  reactance  are  a  maximum.  They 
decrease  with  increasing  positive  slip.  With  increasing 
negative  slip  the  apparent  impedance  and  reactance  decrease 
also,  the  apparent  resistance  decreases  to  zero  and  then  in- 
creases again  in  negative  direction  as  shown  in  Fig.  140, 


308 


EL  E  C  TRIG  A  L   AA  GINEERINL 


which  gives  the  apparent  impedance,  resistance,  and  reactance 
of  the  machine  shown  in  Figs.  125  and  126,  etc.,  with  the 
speed  as  abscissae. 


EFFECTIVE   IMPEDANCE 
OF  THREE-'PHAS 


Fig.  140. 

The  cause  is,  that  energy  current  flows  in  opposition  to 
the  terminal  voltage  above  synchronism,  and  thereby  the  in- 
duction machine  behaves  as  an  impedance  of  negative 
resistance,  that  is,  adding  an  energy  E.M.F.  into  the  circuit, 
proportional  to  the  current. 

As  may  be  seen  herefrom,  the  induction  machine  when 


INDUCTION-  MACHINES.  309 

inserted  in  series  in  an  alternating  current  circuit  can  be 
used  as  a  booster ;  that  is,  as  an  apparatus  to  produce  and 
insert  in  the  circuit  an  E.M.F.  proportional  to  the  current, 
and  the  amount  of  the  boosting  effect  can  be  varied  by 
varying  the  speed,  that  is,  the  slip  at  which  the  induction 
machine  is  revolving.  Above  synchronism  the  induction 
machine  boosts ;  that  is,  raises  the  voltage  ;  below  synchron- 
ism it  lowers  the  voltage  ;  in  either  case  also  adding  an  out- 
of -phase  E.M.F.  due  to  its  reactance.  The  greater  the  slip, 
either  positive  or  negative,  the  less  is  the  apparent  resist- 
ance, positive  or  negative,  of  the  induction  machine. 

The  effect  of  resistance  inserted  in  the  secondary  of  the 
induction  booster  is  similar  to  that  in  the  other  applications 
of  the  induction  machine ;  that  is,  it  increases  the  slip 
required  for  a  certain  value  of  apparent  resistance,  thereby 
lowering  the  efficiency  of  the  apparatus,  but  at  the  same 
time  making  it  less  dependent  upon  minor  variations  of  speed ; 
that  is,  requires  a  lesser  constancy  of  slip,  and  thus  of  speed 
and  frequency,  to  give  a  steady  boosting  effect. 

VI.   Phase  Converter. 

It  may  be  seen  from  the  preceding,  that  the  induction 
machine  can  operate  equally  well  as  motor,  below  synchron- 
ism, and  as  generator,  above  synchronism. 

In  the  single-phase  induction  machine,  the  motor  or 
generator  action  occurs  in  one  primary  circuit  only,  but  in 
the  direction  in  quadrature  to  the  primary  circuit  a  mere 
magnetizing  current  flows,  either  in  the  armature,  in  the 
single-phase  motor  proper,  or  in  an  auxiliary  field-circuit,  in 
the  monocylic  motor. 

The  motor  and  generator  action  can  occur,  however, 
simultaneously  in  the  same  machine,  some  of  the  primary 
circuits  acting  as  motor,  others  as  generator  circuits.  Thus, 
if  one  of  the  two  circuits  of  a  quarter-phase  induction  machine 
is  connected  to  a  single-phase  system,  in  the  second  circuit 


310  ELECTRICAL   ENGINEERING. 

an  E.M.F.  is  induced  in  quadrature  with  and  equal  to  the 
induced  E.M.F.  in  the  first  circuit  ;  and  this  E.M.F.  can  thus 
be  utilized  to  generate  currents  which,  with  currents  taken 
from  the  primary  single-phase  mains,  give  a  quarter-phase 
system.  Or,  in  a  three-phase  motor  connected  with  two 
of  its  terminals  to  a  single-phase  system,  from  the  third 
terminal,  an  E.M.F.  can  be  derived  which,  with  the  single- 
phase  system  feeding  the  induction  machine,  combines  to  a 
three-phase  system.  The  induction  machine  in  this  applica- 
tion represents  a  phase  converter. 

The  phase  converter  obviously  combines  the  features  of  a 
single-phase  induction  motor  with  those  of  a  double  trans- 
former, transformation  occurring  from  the  primary  or  motor 
circuit  to  the  secondary  or  armature,  and  from  the  secondary 
to  the  tertiary  or  generator  circuit. 

Thus,  in  a  quarter-phase  motor  connected  to  single-phase 
mains  with  one  of  its  circuits, 

If, 

YQ=  g  -\-  jb  =  primary  polyphase  exciting  admittance, 
Z0=  r0  —  J'XQ=  self-inductive  impedance  per  primary  or  ter- 
tiary circuit, 

Zl  =  r^  —  jx^  =  resultant    single-phase    self -inductive  impe- 
dance of  secondary  circuits. 

Let 

e  =  E.M.F.  induced  by  mutual  flux,  and 
Z  =  r  —  jx  =  impedance   of  external   circuit  supplied   by 
the  phase  converter  as  generator  of  second  phase. 

We  then  have, 

/  =  =  current  of  second  phase  generated  by  phase 

&  ~r  ^o 
converter, 

E  —  IZ  =  — —  =  -  — —  =  terminal  voltage  at  genera- 

~*~    °       1  _j_  _° 
Z 
tor  circuit  of  phase  converter. 


INDUCTION  MACHINES.  311 

The  current  in  the  secondary  of  the  phase  converter  is 
then, 


where 

/=  load  current  =  „> 

Z     -f-  ZQ 

/'  =  e  Y  =  exciting  current  of  quadrature  magnetic  flux, 
/'  =  -       :  —  =  current  required  to  revolve  the  machine, 

7\  -JSX^ 

and  the  primary  current  is, 

/0  =  /l  +  /'> 

where 

/'  =  eYQ=  exciting  current  of  main  magnetic  flux. 

From  these  currents  the  E.M.F.'s  are  derived  in  a  similar 
manner  as  in  the  induction  motor  or  generator. 

Due  to  the  internal  losses  in  the  phase  converter,  the 
E.M.F.  of  the  two  circuits,  the  motor  and  generator  circuits, 
are  practically  in  quadrature  with  each  other  and  equal  only 
at  no  load,  but  shift  out  of  phase  and  become  more  unequal 
with  increase  of  load,  the  unbalancing  depending  upon  the 
constants  of  the  phase  converter. 

It  is  obvious  that  the  induction  machine  is  used  as  phase 
converter  only  to  change  single-phase  to  polyphase,  since  a 
change  from  one  polyphase  system  to  another  polyphase 
system  can  be  effected  by  stationary  transformers.  A 
change  from  single-phase  to  polyphase,  however,  requires  a 
storage  of  energy,  since  the  power  arrives  as  single-phase 
pulsating,  and  leaves  as  steady  polyphase  flow,  and  the 
momentum  of  the  revolving  phase  converter  secondary 
stores  and  returns  the  energy. 

With  increasing  load  on  the  generator  circuit  of  the  phase 
converter  its  slip  increases,  but  less  than  with  the  same  load 
as  mechanical  output  from  the  machine  as  induction  motor. 

An  application  of  the  phase  converter  is  made  in  single- 
phase  motors  by  closing  the  tertiary  or  generator  circuit  by 
a  condenser  of  suitable  capacity,  thereby  generating  the  ex- 
citing current  of  the  motor  in  the  tertiary  circuit. 


312  ELECTRICAL   ENGINEERING. 

The  primary  circuit  is  thereby  relieved  of  the  exciting 
current  of  the  motor,  the  efficiency  essentially  increased,  and 
the  power  factor  of  the  single-phase  motor  with  condenser  in 
tertiary  circuit  becomes  practically  unity  over  the  whole 
range  of  load.  At  the  same  time,  since  the  condenser  cur- 
rent is  derived  by  double  transformation  in  the  multitooth 
structure  of  the  induction  machine,  which  has  a  practically 
uniform  magnetic  field,  irrespective  of  the  shape  of  the  pri- 
mary impressed  E.M.F.  wave,  the  application  of  the  condenser 
becomes  feasible  irrespective  of  the  wave  shape  of  the  gen- 
erator. 

Usually  the  tertiary  circuit  in  this  case  is  arranged  on  an 
angle  of  60°  with  the  primary  circuit,  and  in  starting  a  power- 
ful torque  thereby  developed,  with  a  torque  efficiency  supe- 
rior to  any  other  single  phase  motor  starting  device,  and 
when  combined  with  inductive  reactance  in  a  second  ter- 
tiary circuit,  the  apparent  starting  torque  efficiency  can  be 
made  even  to  exceed  that  of  the  polyphase  induction  motor. 

Further  discussion  hereof,  see  A.  I.  E.  E.  Transactions, 
1900,  p.  37. 

VH  Frequency  Converter  or  General  Alternating   Current 
Transformer. 

The  E.M.F.'s  induced  in  the  secondary  of  the  induction 
machine  are  of  the  frequency  of  slip,  that  is,  synchronism 
minus  speed,  thus  of  lower  frequency  than  the  impressed 
E.M.F.  in  the  range  from  standstill  to  double  synchronism  ; 
of  higher  frequency  outside  of  this  range. 

Thus,  by  opening  the  secondary  circuits  oj^the  induction 
machine,  and  connecting  them  to  an  external  or  consumer's 
circuit,  the  induction  machine  can  be  used  to  transform  from 
one  frequency  to  another,  as  frequency  converter. 

It  lowers  the  frequency  with  the  secondary  running  at  a 
speed  between  standstill  and  double  synchronism,  and  raises 
the  frequency  with  the  secondary  either  driven  backward  or 
above  double  synchronism. 


INDUCTION  MACHINES.  313 

Obviously,  the  frequency  converter  can  at  the  same  time 
change  the  E.M.F.,  by  using  a  suitable  number  of  primary 
and  secondary  turns,  and  can  change  the  phases  of  the  system 
by  having  a  secondary  wound  for  a  different  number  of 
phases  than  the  primary,  as,  for  instance,  convert  from  three- 
phase  6000-volts  25-cycle,  to  quarter-phase  2500-volts  62.5- 
cycles. 

Thus,  a  frequency  converter  can  be  called  a  "  General 
Alterating  Current  Transformer." 

For  its  theoretical  discussion  and  calculation,  see  "Theory 
and  Calculation  of  Alternate  Current  Phenomena." 

The  action  and  the  equations  of  the  general  alternating 
current  transformer  or  frequency  converter  are  essentially 
those  of  the  stationary  alternating  current  transformer,  ex- 
cept that  the  ratio  of  secondary  to  primary  induced  E.M.F. 
is  not  the  ratio  of  turns,  but  the  ratio  of  the  product  of 
turns  and  frequency,  while  the  ratio  of  secondary  current 
and  primary  load  current  (that  is,  total  primary  current 
minus  primary  exciting  current)  is  the  inverse  ratio  of 
turns. 

The  ratio  of  the  products  of  induced  E.M.F.  and  current, 
that  is,  the  ratio  of  electric  power  generated  in  the  secondary, 
to  electric  power  consumed  in  the  primary  (less  excitation), 
is  thus  not  unity,  but  is  the  ratio  of  secondary  to  primary 
frequency. 

Hence,  when  lowering  the  frequency  with  the  secondary 
revolving  at  a  speed  between  standstill  and  synchronism, 
the  secondary  output  is  less  than  the  primary  input,  and  the 
difference  is  transformed  in  mechanical  work ;  that  is,  the 
machine  acts  at  the  same  time  as  induction  motor,  and  when 
used  in  this  manner  is  usually  connected  to  a  synchronous 
or  induction  generator  feeding  preferably  into  the  secondary 
circuit  (to  avoid  double  transformation  of  its  output)  which 
transforms  the  mechanical  power  of  the  frequency  converter 
into  electrical  power. 

When  raising  the  frequency  by  backward  rotation,  the 


314  ELECTRICAL   ENGINEERING. 

secondary  output  is  greater  than  the  primary  input  (or  rather 
the  electric  power  generated  in  the  secondary  greater  than 
the  primary  power  consumed  by  the  induced  E.M.F.)  and  the 
difference  is  to  be  supplied  by  mechanical  power  by  driving 
the  frequency  changer  backward  by  synchronous  or  induc- 
tion motor,  preferably  connected  to  the  primary  circuit,  or 
by  any  other  motor  device. 

Above  synchronism  the  ratio  of  secondary  output  to 
primary  input  becomes  negative ;  that  is,  the  induction 
machine  generates  power  in  the  primary  as  well  as  the  sec- 
ondary, the  primary  power  at  the  impressed  frequency,  the 
secondary  power  at  the  frequency  of  slip,  and  thus  requires 
mechanical  driving  power. 

The  secondary  power  and  frequency  are  less  than  the 
primary  below  double  synchronism,  more  above  double  syn- 
chronism. 

As  far  as  its  transformer  action  is  concerned,  the  fre- 
quency converter  is  an  open  magnetic  circuit  transformer ; 
that  is,  a  transformer  of  relatively  high  magnetizing  current. 
It  combines  therewith,  however,  the  action  of  an  induction 
motor  or  generator.  Excluding  the  case  of  over-synchro- 
nous rotation,  as  of  lesser  importance,  it  is  approximately 
(that  is,  neglecting  internal  losses)  electrical  input  H-  elec- 
trical output  -t-  mechanical  output  =  primary  frequency  -s- 
secondary  frequency  -4-  speed  or  primary  minus  secondary 
frequency. 

That  is,  the  mechanical  output  is  negative  when  increas- 
ing the  frequency  by  backward  rotation. 

Such  frequency  converters  are  to  a  certain  extent  in 
commercial  use,  and  have  the  advantage  over  the  motor- 
generator  plant  by  requiring  an  amount  of  apparatus  equal 
only  to  the  output,  while  the  motor-generator  set  requires 
machinery  equal  to  twice  the  output. 

An  application  of  the  frequency  converter  when  lower- 
ing the  frequency  is  made  in  concatenation  or  tandem  control 
of  induction  machines,  as  described  in  the  next  section.  In 


INDUCTION  MACHINES.  315 

this  case,  the  first  motor  (or  all  the  motors  except  the  last  of 
the  series)  are  in  reality  frequency  converters. 

VIII.   Concatenation  of  Induction  Motors. 

In  the  secondary  of  the  induction  motor  an  F.M.F.  is 
induced  of  the  frequency  of  slip.  Thus  connecting  the 
secondary  circuit  of  the  induction  motor  to  the  primary  of  a 
second  induction  motor,  the  latter  is  fed  by  a  frequency 
equal  to  the  slip  of  the  first  motor,  and  reaches  its  synchro- 
nism at  the  frequency  of  slip  of  the  first  motor,  the  first 
motor  then  acting  as  frequency  converter  for  the  second 
motor. 

If,  then,  the  two  induction  motors  are  rigidly  connected 
together  and  thus  caused  to  revolve  at  the  same  speed,  the 
speed  of  the  second  motor,  which  is  the  slip  s  of  the  first 
motor  at  no  load,  equals  the  speed  of  the  first  motor 
s  =  1  —  s,  and  thus  s  =  .5.  That  is,  a  pair  of  induction 
motors  connected  this  way  in  tandem  or  in  concatenation, 
that  is,  "chain  connection,"  as  commonly  called,  tends  to 
approach  ^  =  .5,  or  half  synchronism  at  no  load,  slipping 
below  this  speed  under  load.  That  is,  concatenation  of  two 
motors  reduces  their  synchronous  speed  to  one-half,  and  thus 
offers  a  means  to  operate  induction  motors  at  one-half  speed. 

In  general,  if  a  number  of  induction  machines  are  con- 
nected in  tandem  ;  that  is,  the  secondary  of  each  motor 
feeding  the  primary  of  the  next  motor,  the  secondary  of  the 
last  motor  being  short  circuited,  the  sum  of  the  speeds  of  all 
motors  tends  towards  synchronism,  and  with  all  motors  con- 
nected together  so  as  to  revolve  at  the  same  speed,  the 

system  operates  at  -  synchronous  speed,  when  n  =  number 

of  motors. 

Assuming  the  ratio  of  turns  of  primary  and  secondary  as 
1:1,  with  two  induction  motors  in  concatenation  at  standstill, 
the  frequency  and  the  E.M.F.  impressed  upon  the  r,econd 
motor,  neglecting  the  drop  of  E.M.F.  in  the  internal  impe- 


316  ELECTRICAL   ENGINEERING. 

dance  of  the  first  motor,  equal  that  of  the  first  motor.  With 
increasing  speed,  the  frequency  and  the  E.M.F.  impressed 
upon  the  second  motor  decrease  proportionally  to  each 
other,  and  thus  the  magnetic  flux  and  the  magnetic  density 
in  the  second  motor,  and  its  exciting  current,  remain  constant 
and  equal  to  that  of  the  first  motor,  neglecting  internal  losses. 
That  is,  when  connected  in  concatenation  the  magnetic  den- 
sity, current  input,  etc.,  and  thus  the  torque  developed  by  the 
second  motor,  are  approximately  equal  to  that  of  the  first 
motor,  being  less  only  due  to  the  internal  losses  in  the  first 
motor. 

Hence,  the  motors  in  concatenation  share  the  work  in 
approximately  equal  portions,  and  the  second  motor  utilizes 
the  power  which  without  the  use  of  a  second  motor  at  less 
than  one-half  synchronous  speed  would  have  to  be  wasted 
in  the  secondary  resistance.  That  is,  theoretically  concate- 
nation doubles  the  torque  and  output  for  a  given  current,  or 
power  input  into  the  motor  system.  In  reality  the  gain  is 
somewhat  less,  due  to  the  second  motor  not  being  quite 
equal  to  a  non-inductive  resistance  for  the  secondary  of  the 
first  motor,  and  due  to  the  drop  of  voltage  in  the  internal 
impedance  of  the  first  motor,  etc. 

At  one-half  synchronism,  that  is,  the  limiting  speed  of  the 
concatenated  couple,  the  current  input  in  the  first  motor 
equals  its  exciting  current  plus  the  transformed  exciting  cur- 
rent of  the  second  motor,  that  is,  equals  twice  the  exciting 
current. 

Hence,  comparing  the  concatenated  couple  with  a  single 
motor,  the  primary  exciting  admittance  is  doubled.  The 
total  impedance,  primary  plus  secondary,  is  that  of  both 
motors,  that  is  doubled  also,  and  the  characteristic  constant 
of  the  concatenated  couple  is  thus  four  times  that  of  a  single 
motor,  but  the  speed  reduced  to  one-half. 

Comparing  the  concatenated  couple  with  a  single  motor 
rewound  for  twice  the  number  of  poles,  that  is,  one-half  speed 
also,  such  rewinding  does  not  change  the  impedance,  but 


INDUCTION  MACHINES. 


317 


quadruples  the  admittance,  since  one-half  as  many  turns  per 
pole  have  to  produce  the  same  flux  in  one-half  the  pole  arc, 


Fig.  141. 

that  is,  with  twice  the  density.  Thus  the  characteristic  con- 
stant is  increased  four-fold  also.  It  follows  herefrom  that 
the  characteristic  constant  of  the  concatenated  couple  is 
that  of  one  motor  rewound  for  twice  the  number  of  poles. 


318 


ELECTRICAL   ENGINEERING. 


The  slip  under  load,  however,  is  less  in  the  concatenated 
couple  than  in  the  motor  with  twice  the  number  of  poles, 
being  due  to  only  one-quarter  the  internal  impedance,  the 
secondary  impedance  of  the  second  motor  only,  and  thus 
the  efficiency  is  higher. 

Two  motors  coupled  in  concatenation  are  in  the  range 
from  standstill  to  one-half  synchronism,  approximately  equiv- 
alent to  one  motor  of  twice  the  admittance,  three  times  the 


;6  -7 


Fig.  142. 

primary  impedance,  and  the  same  secondary  impedance  as 
each  of  the  two  motors,  or  more  exactly  2.8  times  the 
primary  and  1.2  times  the  secondary  impedance  of  one 
motor.  Such  a  motor  is  called  the  "  Equivalent  Motor." 

The  calculation  of  the  characteristic  curve  of  the  concat- 
enated motor  system  is  similar  to,  but  more  complex  than, 
that  of  the  single  motor.  Starting  from  the  induced  E.M.F. 
e  of  the  second  motor,  reduced  to  full  frequency,  we  work  up 
to  the  impressed  E.M.F.  of  the  first  motor  e^  by  taking  due 
consideration  of  the  proper  frequencies  of  the  different  cir- 


IND  UC  TION  MA  CHINES. 


319 


cults.  Herefor  the  reader  must  be  referred  to  "  Theory  and 
Calculation  of  Alternate  Current  Phenomena,"  3d  edition. 

The  load  curves  of  the  pair  of  three-phase  motors  of  the 
same  constants  as  the  motor  in  Figs.  125  and  126  are  given 
in  Fig.  141,  the  complete  speed  curve  in  Fig.  142. 

Fig.  141  shows  the  load  curve  of  the  total  couple,  of  the 
two  individual  motors,  and  of  the  equivalent  motor. 

As  seen  from  the  speed  curve,  the  torque  from  stand- 
still to  one-half  synchronism  has  the  same  shape  as  the 
torque  curve  of  a  single  motor  between  standstill  and  syn- 
chronism. At  one-half  synchronism  the  torque  reverses  and 


•6000 
-4000 
2000 
0 

—      ~. 

CC 

s. 

)NCAfENAtlON   JOF   I.NJDUCT 
SPEED  CUPVES 

ION   NIOTO 

*S. 

J 

N.  z 

0=Z=.1-.3J    Y0=.C 
RES.   I'll  SECONDARY 

1-K1 

J 

d 

§—   - 

**x 

V 

OF  SEC 

OND  M 

OTOR. 

CO 

i 

co 

5 

-—  —  -*^ 

1 

co 

-2000 
-4000 

-60oo1 

v 

,'/ 

s*~ 

v 

/ 

/ 

.0       . 

9 

3 

7 

5 

N. 

>             . 

*s 

\ 

? 

2 

1 

D 

Fig.  143. 

becomes  negative.  It  reverses  again  at  about  f  synchronism, 
and  is  positive  between  about  f  synchronism  and  synchro- 
nism, zero  at  synchronism,  and  negative  beyond  synchronism. 

Thus,  with  a  concatenated  couple,  two  ranges  of  positive 
torque  and  power  as  induction  motor  exist,  one  from  stand- 
still to  half  synchronism,  the  other  from  about  f  synchronism 
to  synchronism. 

In  the  ranges  from  J  synchronism  to  about  ^  synchro- 
nism, and  beyond  synchronism,  the  torque  is  negative,  that 
is,  the  couple  acts  as  generator. 

The  insertion  of  resistance  in  the  secondary  of  the 
second  motor  has  in  the  range  from  standstill  to  half  syn- 
chronism the  same  effect  as  in  a  single  induction  motor,  that 
is,  shifts  the  maximum  torque  point  towards  lower  speed 


320  ELECTRICAL   ENGINEERING. 

without  changing  its  value.  Beyond  half-synchronism,  how- 
ever, resistance  in  the  secondary  lengthens  the  generator 
part  of  the  curve,  and  makes  the  second  motor  part  of  the 
curve  more  or  less  disappear,  as  seen  in  Fig.  143,  which 
gives  the  speed  curves  of  the  same  motor  as  Fig.  142,  with 
resistance  in  circuit  in  the  secondary  of  the  second  motor. 

The  main  advantages  of  concatenation  are  obviously  the 
ability  of  operating  at  two  different  speeds,  the  increased 
torque  and  power  efficiency  below  half-speed,  and  the  gene- 
rator or  braking  action  between  half-speed  and  synchronism. 


INDEX. 


PAGE 

Acceleration  with  starting   device,  sin- 
gle-phase induction  motor   .     .     ,  295 

Admittance 105 

absolute,  of  induction  motor    .     .     .  275 

primary,  of  transformer 85 

and  impedance 103 

Algebraic  calculation  of  transformer  .     .  75 
Alternating   current  circuits,  self-induc- 
tion      29 

currents,  effect 38 

current      generator,      characteristic 

curves     .          139 

current  induction,  formula    ....  16 

current  transformer 69 

E.M.F. 13 

wave 30 

Alternators 123 

Ampere 2 

Ampere-turn 2 

Ampere-turns  per  cm 3 

Angle  of  hysteretic  lead 50 

of  lag 34 

Armature  current  and   heating   of  con- 
verters     * 228 

reaction,  commutating  machines  .     .  187 

reaction  of  converters 236 

reaction  of  synchronous  machines     .  129 
resistance    in    polyphase    induction 

motor 277 

winding 1G5 

Apparent  efficiency .  270 

power  efficiency 270 

reactance 52 

torque  efficiency 270 

Arc  lamp 125 

light  machines 124 

Asynchronous,  see  induction. 

Auto-transformer 125 

See  compensator. 
Auxiliary  magnetic  flux,  single-phase  in- 

'-  •      duction  motor 291 

Average  induced  E.M.F 12 

Backward  rotation  of  induction  machine ,  297 
Balancing  effect   of   polyphase  synchro- 
nous motors .  152 


Battery,  primary  and  secondary      .     .     .  125 

Biphase,  see  quarter-phase. 

Bipolar  machines,  commutating     .     .     .  163 

Booster 123,125 

induction 307 

Brushes,  angle  of  lead 181 

Brush  arc  machine 124 


Calculation  of  polyphase  induction  mo- 
tor        266 

Capacity  and  condensers 55 

reactance 57 

of  transmission  line 57 

in  induction  motor  secondary  ...  57 
Characteristic  constant  of  induction  mo- 
tor        276 

constant  of    single-phase   induction 

motor 286 

curves  of  alternating  current  genera- 
tor       139 

curves  of  commutating  machines  .     .  192 

curves  of  synchronous  motors  .     .     .  143 

magnetic 8 

Charge  of  condenser 55 

Charging  current  of  condenser  .     .     .     .  56, 

current  of  transmission  line      ...  57 

Choking  coil 125 

Circle  representing  sine   wave  in  polar 

coordinates 41 

Classification  of  electrical  apparatus  .     .  121 

Closed  circuit  winding 168 

Coefficient  of  hysteresis 52 

Commutating  machines     ....      122,  162 

machines,  synchronous 124 

Commutation 193 

of  converter 242 

of  direct  current  converter   ....  259 

Compass  needle 5 

Compensator 125 

with  direct  current  converter    .     .     .  252 

synchronous 123 

Complex  imaginary  quantity,  introduc- 
tion         •     .     .     .  80 

Components,  energy  and  wattless  ...  40 

Compounding  commutating  machines     .  190 

of  converter   .     .  .242 


321 


322 


INDEX. 


PAGE 

Compounding  curve  ot  commutating  ma- 
chine       191 

curves  of  synchronous  generator  .     .     139 
curves  oi  synchronous  motor    .     .     .     144 
ol  transmission  line  for  constant  vol- 
tage     95 

Compound  generator,  commutating    .     .211 

machines,  commutating 163 

motor,  commutating 215 

Concatenation  of  induction  motors      .     .     315 

Condenser 125 

and  capacity 55 

charging  current 56 

Conductance 105 

Conductivity,  magnetic 4 

Constant  current  load  saturation  curve  .  189 
resistance  load  saturation  curve  .  .  190 
speed  induction  generator  ....  299 

Continuous  current,  etc.,  see  direct  cur- 
rent, etc. 

Continuous  E.M.F 13 

Converter 122,  124 

direct  current 123, 251 

frequency 124,  312 

and  induction  generator 304 

and  motor  generator 217 

phase 124 

phase  induction 309 

synchronous 217 

as  synchronous  motor,  etc.  ....    250 

Coordinates,  polar 41 

rectangular 79 

Counter  E.M.F.  of  impedance  ....      42 

of  resistance       ...         ...     32, 42 

of  self-induction 31,  42 

Cross-currents  in   parallel   operation   of 

alternators  .......      155,  157 

Cross-magnetization,  single-phase  induc- 
tion motor 283 

Cumulative  compounding      ....  163,  215 

Current,  electric 2,9 

starting  of 22 

stopping  of 24 

Cylinder  winding,  see  drum  winding  .     .     164 

Demagnetization  curve,  commutating 

machine 205 

of  field  pole,  commutating  machine  .     179 
Demagnetizing  and   magnetizing  arma- 
ture reaction  of  converter     .     .     .    240 
armature  reaction,  synchronous  ma- 
chines      130 

Diamagnetic  material 4 

Dielectric  hysteresis  current  of  condenser,      57 
Differential  compounding      ....  163,  215 
Diphase,  see  quarter-phase. 
Direct  current  converter 123,  251 


PAGE 

Direct  current  induction,  formula  .     .     ,  13 

self-induction 22 

starting  of 22 

stopping  ot ,  24 

Distorting    armature    reaction    of    con- 
verter       240 

of  synchronous  machines      ....  130 
Distortion  of    field,   commutating    ma- 
chine        179 

Distribution  of 'magnetic  flux,  commutat- 
ing machines   177 

Division  of  load  in  parallel  operation  ot 

alternators 156 

Double  current  generators     ....  124,  248 

Double  reentrant  armature  winding    .     .  169 

Double  spiral  armature  winding     .     .     .  169 

Drum  winding 164 

Dynamotors 123 

Eddy  currents,  effective  resistance     .     .  53 

in  pole  face 185 

Effect  and  effective  values 15 

of  armature  resistance  and  starting, 

polyphase  induction  motor    .     .     .  277 

of  alternating  currents 38 

of  saturation  on  magnetic  distribu- 
tion       181 

of  slots  on  magnetic  flux      ....  184 

transmission  line 41 

Effective  alternating  wave 16 

impedance 53 

reactance 52,  104 

resistance .     .     .  52.  104 

resistance  and  hysteresis      ....  48 

values  and  effect 15 

Efficiency 270 

and  losses,  commutating  machines   .  193 

and  losses  in  synchronous  machines  .  150 

curves  of  synchronous  machines  .     .  151 

of  induction  motor 272 

Electric  current 9 

Electrolytic  apparatus  ;......  122 

cell 12& 

Electromotive  force,  see  E.M.F. 

Electrostatic  apparatus 122 

E.M.F 9 

and  magnetism 9 

average  induced 12 

induction  of 11 

instantaneous  induced 12 

maximum  induced 12 

of  self-induction »9 

Energy  cross-currents  of  alternators    .     .  156 

current 40 

E.M.F 39 

Engine  regulation  with  alternators      .     .  158 

Equalizers  with  compounded  alternators  .  161 


INDEX. 


323 


PAGE 

Equivalent  induction  motor,  concatena- 
tion         ;     ....  317 

sine  waves :     '.  112 

Excitation,  field 11 

single-phase  induction  motor   .     .     .  283 

Exciter  of  induction  generator    ....  306 
Exciting  current  of  magnetic  circuit    .     49,  50 

current,  primary,  of  transformer  .     .  71 

current,  single-phase  induction  mo- 
tor        282 

External  characteristic,  series  generator  .  208 

characteristic,  shunt  generator      .     .  206 

Face  commutator  machines 164 

Farad 55 

Field    characteristic,    commutating    ma- 
chine   192 

characterise   of    synchronous  gen- 
erator       140 

excitation 11 

intensity,  magnetic 1 

Fluctuating  cross-currents  in  parallel  op- 
eration of  alternators 157 

Flux,  magnetic -1 

Form  factor  of  alternating  wave      ...  16 
Form  factor  in  synchronous  machines     .  126 
Formula   of    alternating  current  induc- 
tion      16 

of  direct  current  induction    ....  13 

induction,  of  synchronous  machines,  126 

Foucault  currents,  effective  resistance     .  53 

Four-phase,  see  quarter-phase. 

Fractional  pitch  armature  winding      .     .  173 

Frequency  of  commutation 195 

converters 124 

Frequency  converter  or  general  alternat- 
ing current  transformers  ....  312 
Friction,  molecular  magnetic      ....  49 

Full  pitch  armature  winding 173 

Furnace,  electric 125 

Generator  action  of  concatenated  couple,  319 

action  of  frequency  converter  .     .     .  314 

Generators,  alternating 123 

alternating     current,     characteristic 

curves 139 

commutating 123 

double  current 124  248 

induction 124  297 

Gramme  winding,  see  ring  winding     .     .  164 

Heater,  electric 125 

Heating  and  armature  currents   of  con- 
verter        228 

of  direct  current  converters      .     .     .  259 

Henry 20 

Hexaphase,  see  six-phase. 


PAGE 

Horizontal  component  of  sine  wave    .     .  80 
intensity     of     terrestrial     magnetic 

field 14 

Hunting  of  synchronous  machine   .     .     .  158 

Hysteresis  coefficient 52 

current 50 

dielectric 57 

and  effective  resistance 48 

loss 52 

magnetic 49 

Hysteretic  effective  resistance    ....  52 

lead  angle 50 

Imaginary  quantity,  introduction  ...  80 

Impedance 34,  104 

absolute,  of  induction  motor    .     .     .  275 

and  admittance  ........  105 

counter  E.M.F  of 42 

effective 53 

effective,  of  induction  machine     .     .  308 

E.M.F.  consumed  by 42 

of  transmission  lines 58 

Incandescent  lamp 125 

Induced  E.M.F.,  average 12 

commutating  machines 175 

instantaneous 12 

maximum 12 

of  synchronous  machines     ....  128 

Inductance 18 

factor,  induction  generator  ....  301 

unit  of 20 

Induction,  alternating  current, formula  .  16 
apparatus,  stationary  ....       122,  125" 

booster 307 

converters,  see  reaction  converters. 

direct  current,  formula 13 

of  E.M.F.'s 11 

formula  of  synchronous  machines      .  126 

generator 297 

generator  and  converter 303 

generator,  constant  speed     ....  299 

generator,  magnetic  saturation     .     .  302 

generator,  power  factor 301 

generator  and  synchronous  machine,  303 

machines 122,  124.  261 

magnetic 4 

motor 65.  124 

motors,  concatenation  of      ....  315 

motor,  single-phase,  starting    .     .     .  108 

mutual  and  self 18 

phase  converter 309 

Inductive   devices,  starting  single-phase 

induction  motor 292 

Inferiority  of  single-phase  induction  mo- 
tor        286 

Instantaneous  induced  E.M.F.      ...  12 

Intensity  of  alternating  wave      ....  43 


324 


INDEX. 


PAGE 

Intensity,  magnetic 2 

JnterJinkages,  magnetic 18 

Inverted  converters 124,246 

Iron-clad  armatures  .     .  .163 


J.,/,  introduction  of 


Lag  angle 34 

Lagging  current  in  converter      ....  242 

Lamp,  incandescent  and  arc 125 

Lap  winding  of  armature 170 

Lead  angle  of  brushes 181 

hysteretic 50 

Leading  current  in  converter      ....  L'42 

Lentz's  law 10 

Lines  of  magnetic  force 1 

Load,  affecting  converter  ratio  ....  226 
curves,   induction    generators,    con- 
stant frequency 298 

characteristic,  series  generator  .  .  209 
characteristic,  series  motor  ....  214 
characteristic,  shunt  generator  .  .  206 
characteristic,  shunt  motor  ....  212 
characteristics  of  synchronous  motor,  145 
characteristic  of  transmission  line  .  89 
current,  primary,  of  transformer  .  .  71 
curves,  induction  generator,  con- 
stant speed 300 

curves,  induction  motor  concatena- 

tion .  319 

curves  of  synchronous  generator  .     .  141 

magnetic  effect  on  field 183 

saturation    curve,  commutating  ma- 
chine        189 

saturation  curve  of  commutating  ma- 
chine         203 

saturation  curve  of  synchronous  ma- 
chine   148 

and  speed  curves,  polyphase  induc- 
tion motor 272 

and  speed  curves  of  single-phase  in- 
duction motor .     .     , 286 

Loss  curves  of  synchronous  machines     .  151 
Losses  and  efficiencyj  commutating  ma- 
chines       193 

ard   efficiency   ot   synchronous  ma- 
chines   150 

Loss  of  energy  in  pole  face 186 

"Magnet  field,  inductance 21 

Magnetic    characteristic    or    saturation 

curve  of  synchronous  machine     .  147 

distribution,  affected  by  saturation    .  181 

energy  current 50 

field 1 

field  of  force 1 

flux  ...  I 


KAGB 

Magnetic  flux,  affected  by  slots      ...  184 
rlux,    auxiliary    quadrature,    single- 
phase  induction  motor      ....  291 
flux  distribution,  commutating   ma- 
chines       177 

flux,  pulsation  in  pole  face  ....  185 

friction       49 

hysteresis 49 

induction 4 

interlinkages 18 

material 4 

reaction 9 

Magnetism  and  electric  currents     ...  1 

and  E.M.F 9 

Magnetization  curves 8 

of  field  pole  commutating  machines  .  179 
Magnetizing  armature  reaction,  synchro- 
nous machines 130 

current 50 

and  demagnetizing  armature  reaction 

of  converter 240 

force 3 

Magneto  machines,  commutating   .     .     .  162 

Magnetomotive  force 2 

Magneto  and  separately  excited  gener- 
ator, commutating  ......  205 

Magnet  pole 1 

Maximum  induced  E.M.F 12 

mf  =  microfarad 55 

mh  =  milhenry 21 

Microfarad 55 

Milhenry 21 

M.M.F.,  see  magnetomotive  force. 

Molecular  magnetic  friction 49 

Monocyclic  devices,  starting  single-phase 

induction  motor 292 

induction  motor 264 

Mono-phase,  see  single-phase. 

Motor  action  of  frequency  converter   .     ,  314 

Motors,  commutating 123 

Motor  generator  and  converter  .     ...  217 

Motor,  induction 124 

synchronous  123, 141 

synchronous,  characteristic  curves    ,  143 
Multiple  armature  winding    .     .     .       164.  165 

drum  winding 165 

reentrant  armature  winding      ...  169 

ring  winding 165 

spiral  armature  winding  ....  169 

Multipolar  machines,  commutating     .     .  163 

Mutual  inductance 18 

Mutual  induction  and  self-induction   .     .  18 

Negative  resistance  of  induction  ma- 
chine.  t 308 

Neutral  conductor  of  three-wire  system, 

and  converter  ..<..-  •  •  251 


INDEX. 


325 


PAGE 

Neutral  point,  line  or  zone, commutating 

machines 178 

Nomenclature  of  electrical  apparatus      .  121 
Nominal  induced  E.M.F.  of  synchronous 

machine 128 

Nonpolar  induction 11 

machine 125 

N-phase  converter     .     .     .     226,233,235,236 
direct  current  converter  .     .     .      252,  259 

Ohm  ....    .,£. 9 

Ohms  law     .    .    v. 103 

Open  circuit  winding 168 

coil  arc  machines 124 

Oscillating    armature    reaction    of   con- 
verter        240 

Over-compounding  of  converter      .     .     .  244 

curve  of  commutating  machine     .     .  192 

of  transmission  lines 98 

Over-synchronous  rotation  of  induction 

machine 298 

Parallel  connection 107 

Parallelogram  of  sine  waves 44 

Parallel  operation  of  alternators    .     .     .  154 
operation  of  alternators,  division  of 

load 156 

operation  of  alternators,  fluctuating 

cross-currents 157 

Permeability 4 

Phase 13 

of  alternating  wave 43 

angles 65 

characteristics  of  synchronous  motor,  146 

control  of  transmission  lines    ...  94 

converters 12  J,  124 

converter,  induction 309 

displacement,     affecting     converter 

ratio       226 

displacement  in  converter,  affecting 

rating 235 

displacement  in  inverted  converter  .  247 
splitting     devices,    starting   single- 
phase  induction  motor     ....  291 

Polar  coordinates 41 

Polarization  cell 125 

Pole  face,  eddy  currents 185 

energy  loss ,186 

pulsation  of  flux 185 

Pole,  magnetic 1 

Polygon  of  sine  waves 44 

Polyphase  induction  machines  ....  124 

induction  motor 264 

induction  motor  on  single-phase  cir- 
cuit       285 

synchronous  machines,  unbalancing  151 

Potential  regulator 125 


PAGE 

Power  efficiency        270 

factor  of   admittance   and  of  impe- 
dance, induction  motor     ....  275 
factor  of  induction  generator   .     .     .  301 

factor  of  reactive  coil 53 

Primary  admittance  of  transformer      .     .  85 

Pulsating  armature  reaction  of  converter  240 

Pulsation  of  magnetism  in  pole  face  .     .  185 

Quarter-phaser,  armature  reaction  .  .  132 
Quarter-phase  converter,  ratio  221,  226,  233, 

235,236 

direct  current  converter  .  .  252,  259 
Quadrature  magnetic  flux,  single-phase 

induction  motor       291 

Racing  of  inverted  converter  ....  247 
Rating  of  converter,  based  on  armature 

heating 228 

of  direct  current  converters      .     .     .  259 
of  synchronous   machines  by  arma- 
ture heating 236 

Ratio  of   E.M.Fs.    and   of  currents  in 

converters 218 

Reactance 31 

capacity 57 

effective 104 

effective  or  apparent 52 

self-inductive 52 

synchronous 129,  136 

of  transmission  line 34 

Reaction,    armature,   commutating    ma- 
chines    .     . 187 

armature,  of  converters 236 

armature,  of  synchronous  machines  129 

converters 243 

magnetic 9 

Reactive  coil 53,  125 

Real  induced  E.  M.  F.  of  synchronous 

machine     .     .     .     , 128 

Rectangular  components 65,  79 

Rectifying  apparatus 122 

machines 124 

Reentrant  armature  winding      ....  169 

Regulation  curve,  commutating  machine  192 

curves  of  synchronous  generator       .'  140 

of  speed  of  alternators     .     .     .     .     .  156 

of  transformer 79 

Regulator,  potential 125 

Reluctance,  magnetic 19 

Repulsion  motor 290 

Resistance 9 

characteristic,  series  generator     .     .  209 

characteristic,  shunt  generator     .     .  208 

commutation 195, 201 

counter  E.  M.  F.  of 32,  42 

curves,  polyphase  induction  motor   .  280 


326 


INDEX. 


PAGE 

Resistance,  effective 52,  104 

effective,  of  eddy  currents    ....  53 

effective,  and  hysteresis 48 

E.  M.  F.  consumed  by    ....     32,  42 

hysteretic 52 

inductance  starting  device  of  single- 
phase  induction  motor       ....  293 
negative,  of  induction  machine     .     .  308 
speed  curves  of  single-phase  induc- 
tion motor       289 

Resistivity 9 

Resultant  of  rectangular  components      .  80 

Ring  winding 164 

Rotary  converter,  see  converter. 
Rotary  transformer,  see  converter. 

Saturation  affecting  magnetic  distribu- 
tion  . 181 

coefficient,  commutating  machine  .  189 
coefficient  of  synchronous  machine  .  148 
curves,  commutating  machines,  188,  202 
curve  or  magnetic  characteristic  of 

synchronous  machine 147 

of  field,  affecting  synchronous   ma- 
chines      149 

Self-inductance 19 

Self-induction  of  alternating  current  cir- 
cuits   29 

of  direct  current  circuits  ....  22 
for  converter  compounding  .  .  .  243 

counter  E.  M.  F.  of 31,  42 

E.  M.  F.  of '.     .     .     .        9 

E.  M.  F.  consumed  by   .     .     .     .     32,42 

and  mutual  induction 18 

of  synchronous  machines  ....  133 
of  transmission  line 17 

Self-inductive  reactance 52 

Separately  excited  machines,  commutat- 
ing      162 

Separately  excited  and  magneto  gener- 
ator, commutating 205 

Series  armature  winding    ....       164,  166 

connection 107 

drum  winding 166 

generator,  commutating  .  .  .  .  ,j;--*208 
machines,  commutating  .  .  ^.v^v"  1,62 

motors /£'   /  .     123. 

motor,  commutating  .     .     .  j'i .    V'  i    vn'' 

Shift  of  brushes '*..  .     .     181 

Shunt  generator,  commutating  .     . '^^-^206- 

machines,  commutating "162 

motors 123 

motor,  commutating  ......    212 

Sine  waves,  equivalent 112 

waves  represented  by  circle      ...      41 

Single  circuit  single-phase  converter  223,  226, 
233,  235,  236 


PAGR 

Single-phase  converter,  ratio  .     220,  223,  226, 

233,  235,  236 

direct  current  converter  .     .     .      252,  259 

induction  machines 124 

induction  motor 264, 281 

induction  motors,  starting  devices  .  290 
Single-phaser,  armature  reaction  .  .  .  132 
Six-phase  converter  .  .  224,  226,  233,  235,  236 
Slip  and  armature  resistance,  induction 

motor 277 

Slots,  affecting  magnetic  flux     ....     184 

Smooth  core  armature 163 

Sparking  of  converter 242 

Sparkless  commutation 200 

Speed  characteristic,  shunt  generator       .     209 
curves,  induction  machine   ....     299 
curves,  induction  motor  concatena- 
tion     319 

curves  of  polyphase  induction  motor    278 

curves,  series  motor 214 

curves,  shunt  motor    ....       212,  213 

of  inverted  converter 247 

and  load  curves,  polyphase  induction 

motor 272 

and  load  curves  of  single-phase  in- 
duction motor     286 

regulation  of  alternators 156 

variation  of  engines  driving  alterna- 
tors     158 

Spiral  armature  winding 169 

Star  connected  three-phaser,  E.M.F.      .      16 

Starting  of  converters 244 

device  and  acceleration,  single-phase 

induction  motor 295 

devices  of  single-phase  induction  mo- 
tors     290 

of  direct  current 22 

of  polyphase  induction  motor      .     .    277 

of  synchronous  motors 153 

torque,  polyphase  induction  motor  .    280 

Stopping  of  current 24 

Strength  of  magnet  pole 2 

Susceptance 105 

Symbolic  calculation  of  transformer   .     .      80 
~>  ;iepresentalion  of  sine  waves     ...      80 

Synchronizing,  converters 245 

Synchronous  cbmmutaling  machines  .     .     124 

'converters- 124,217 

induction  machine 307 

.rriachthes 122,126 

'motor 123,  141 

motor,  characteristic  curves      .     .     .     143 
motor  operated  by  induction  genera- 
tor       303 

motors,  starting 153 

reactance 129,  136 

watts     .  .    269 


INDEX. 


327 


PAGE 

Telephone  line,  mutual  inductance     .     .      21 
Tandem  control  of  induction  motors  .     .    315 
Teeth  of  armature,  saturation     ....     183 
Terminal   voltage    of    synchronous   ma- 
chine       128 

Terrestrial  magnetic  field 5 

Thermopile 125 

Thomson-Houston  arc  machine      .     .     .    124 

Three-phase  armature  winding  ....     168 

converter,  ratio  .     .221,226,233,235,236 

direct  current  converter   .     .     .      252,  259 

machines 127 

Three-phaser,  armature  reaction     .     .     .     132 

E.M.F 16 

Time  constant  of  circuit 23,  34 

Toothed  armatures 163 

Torque  curves,  series  motor 214 

efficiency 270 

single-phase  induction  motor  .     .     .     284 

in  synchronous  watts 269 

Transmission  line  capacity 57 

charging  current 57 

compounding  for  constant  voltage    .      95 

efficiency „     .      41 

impedance      .     .     .     „ 58 

load  characteristic 89 

magnetic  flux 6 

overcompounding .     .      98 

phase  control 45^  68,  88;  94 

reactance 34 

self-induction -.      17 

Transformer 69,  122,  125 

general  alternating  current  ....    312 
rotary,  see  converter, 
wave  shape  of  exciting  current      .     .     114 
Trigonometrical    calculation    of    trans- 
former         .....      72 

Triphase,  see  three-phase. 
Twelve-phase  converter     .    226,  233,  235,  236 
Two-circuit  single-phase  converter,     223,  226, 
233,235,236 


Types  of  commutating  machines 


PAGE 

.    202 


Unbalancing  of  polyphase  synchronous 

machines     .......     .     .  .  151 

Uniphase,  see  single-phase. 

Unipolar  induction   ........  11 

machine     ..........  125 

Unit  capacity   ..........  55 

current  .........    ,    ,  2 

E.M.F  ...........  9 

of  inductance      ........  20 

magnetic  field     ......    ,     .  1 

magnet  pole  .    .     .    ......  1 

resistance  .....    ....'."  9 

Variation  of  the  ratio  of  E.M.F.  's  of 

Vector,  polar  representation  of  alternat- 

ing wave     ......     oo.  43 

Vertical  component  of  sine  wave    ...  80 
Virtual  induced  E.M.F.  of  synchronous 

machine      .    «     .......  128 

Volt     .........    .     .    .     .  9 

Voltage  commutation    ...,.».  196 

Wattless  cross-currents  of  alternators   .  156 

current  .........    .     .  40 

currents  and   compounding  of  con- 

verter     ......     ....  242 

E.M.F  ........     ...  39 

Wave,  alternating    ........  30 

shape,  affecting  converter  ratio    .    .  226 
shapes   of    converter  armature  cur- 

rents .     .     .........  229 

shape  of  direct  current  converter  ar- 

mature currents  .......  255 

shape  of  synchronous  machines    .     .  127 

winding  of  armature    ......  170 

Winding,  armature  ........  165 

Zero  vector,  choice  of  .......  45 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN     INITIAL     FINE     OF     25     CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


OCT   10  193; 


SEC.  c/u.  KW  28  78 

J  AN  1 2  1979 


MAR  11 


APR    18  1935 


JAM  25  I! 


4 

OS.VIS 

INTER-LIBRARY 

LOAN 
OCT    2  1967 

USE 


CIR.    JAN  9      1979 


NOV  2  fflSH 


LD  21-50m-8,-32 


• 


